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Friable values of polynomials Greg Martin University of British - PowerPoint PPT Presentation

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Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Friable values of polynomials Greg Martin University of British Columbia PIMS/SFU/UBC Number Theory Seminar March 3,

  1. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Friable numbers among values of polynomials Definition Ψ( F ; x , y ) = # { 1 ≤ n ≤ x : p | F ( n ) = ⇒ p ≤ y } is the number of integers n up to x such that all the prime factors of F ( n ) are all at most y . When F ( x ) is a linear polynomial (friable numbers in arithmetic progressions), we have the same asymptotic � log x � formula Ψ( F ; x , y ) ∼ x ρ . log y Knowing the size of Ψ( F ; x , y ) has applications to analyzing the running time of modern factoring algorithms (quadratic sieve, number field sieve). Fundamental question Are two arithmetic properties (in this case, friability and being the value of a polynomial) independent? Friable values of polynomials Greg Martin

  2. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment How friable can values of special polynomials be? For binomials, there’s a nice trick which yields: Theorem (Schinzel, 1967) For any nonzero integers A and B , any positive integer d , and any ε > 0 , there are infinitely many numbers n for which An d + B is n ε -friable. Balog and Wooley (1998), building on an idea of Eggleton and Selfridge, extended this result to products of binomials L ( A j n d j + B j ) , � j = 1 which includes products of linear polynomials. Friable values of polynomials Greg Martin

  3. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment How friable can values of special polynomials be? For binomials, there’s a nice trick which yields: Theorem (Schinzel, 1967) For any nonzero integers A and B , any positive integer d , and any ε > 0 , there are infinitely many numbers n for which An d + B is n ε -friable. Balog and Wooley (1998), building on an idea of Eggleton and Selfridge, extended this result to products of binomials L ( A j n d j + B j ) , � j = 1 which includes products of linear polynomials. Friable values of polynomials Greg Martin

  4. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Proof for an explicit binomial Example Let’s show that for any ε > 0 , there are infinitely many numbers n for which F ( n ) = 3 n 5 + 7 is n ε -friable. Define n k = 3 8 k − 1 7 2 k . Then F ( n k ) = 3 5 ( 8 k − 1 )+ 1 7 5 ( 2 k ) + 7 = − 7 ( − 3 4 7 ) 10 k − 1 − 1 � � factors into values of cyclotomic polynomials: � Φ m ( − 3 4 7 ) . F ( n k ) = − 7 m | ( 10 k − 1 ) � x − e 2 π ir / m � � Φ m ( x ) = 1 ≤ r ≤ m ( r , m )= 1 Φ m has integer coefficients and degree φ ( m ) Friable values of polynomials Greg Martin

  5. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Proof for an explicit binomial Example Let’s show that for any ε > 0 , there are infinitely many numbers n for which F ( n ) = 3 n 5 + 7 is n ε -friable. Define n k = 3 8 k − 1 7 2 k . Then F ( n k ) = 3 5 ( 8 k − 1 )+ 1 7 5 ( 2 k ) + 7 = − 7 ( − 3 4 7 ) 10 k − 1 − 1 � � factors into values of cyclotomic polynomials: � Φ m ( − 3 4 7 ) . F ( n k ) = − 7 m | ( 10 k − 1 ) � x − e 2 π ir / m � � Φ m ( x ) = 1 ≤ r ≤ m ( r , m )= 1 Φ m has integer coefficients and degree φ ( m ) Friable values of polynomials Greg Martin

  6. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Proof for an explicit binomial Example Let’s show that for any ε > 0 , there are infinitely many numbers n for which F ( n ) = 3 n 5 + 7 is n ε -friable. Define n k = 3 8 k − 1 7 2 k . Then F ( n k ) = 3 5 ( 8 k − 1 )+ 1 7 5 ( 2 k ) + 7 = − 7 ( − 3 4 7 ) 10 k − 1 − 1 � � factors into values of cyclotomic polynomials: � Φ m ( − 3 4 7 ) . F ( n k ) = − 7 m | ( 10 k − 1 ) � x − e 2 π ir / m � � Φ m ( x ) = 1 ≤ r ≤ m ( r , m )= 1 Φ m has integer coefficients and degree φ ( m ) Friable values of polynomials Greg Martin

  7. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Proof for an explicit binomial Example Let’s show that for any ε > 0 , there are infinitely many numbers n for which F ( n ) = 3 n 5 + 7 is n ε -friable. Define n k = 3 8 k − 1 7 2 k . Then F ( n k ) = 3 5 ( 8 k − 1 )+ 1 7 5 ( 2 k ) + 7 = − 7 ( − 3 4 7 ) 10 k − 1 − 1 � � factors into values of cyclotomic polynomials: � Φ m ( − 3 4 7 ) . F ( n k ) = − 7 m | ( 10 k − 1 ) � x − e 2 π ir / m � � Φ m ( x ) = 1 ≤ r ≤ m ( r , m )= 1 Φ m has integer coefficients and degree φ ( m ) Friable values of polynomials Greg Martin

  8. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Proof for an explicit binomial Example Let’s show that for any ε > 0 , there are infinitely many numbers n for which F ( n ) = 3 n 5 + 7 is n ε -friable. Define n k = 3 8 k − 1 7 2 k . Then F ( n k ) = 3 5 ( 8 k − 1 )+ 1 7 5 ( 2 k ) + 7 = − 7 ( − 3 4 7 ) 10 k − 1 − 1 � � factors into values of cyclotomic polynomials: � Φ m ( − 3 4 7 ) . F ( n k ) = − 7 m | ( 10 k − 1 ) � x − e 2 π ir / m � � Φ m ( x ) = 1 ≤ r ≤ m ( r , m )= 1 Φ m has integer coefficients and degree φ ( m ) Friable values of polynomials Greg Martin

  9. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment From the last slide F ( n ) = 3 n 5 + 7 � Φ m ( − 3 4 7 ) F ( n k ) = − 7 n k = 3 8 k − 1 7 2 k m | ( 10 k − 1 ) � Φ m ( − 3 4 7 ) � � the primes dividing F ( n k ) are at most max � m | ( 10 k − 1 ) each Φ m ( x ) is roughly | x | φ ( m ) ≤ | x | φ ( 10 k − 1 ) n k is roughly ( 3 4 7 ) 4 k , but the largest prime factor of F ( n k ) is bounded by roughly ( 3 4 7 ) φ ( 10 k − 1 ) there are infinitely many k with φ ( 10 k − 1 ) / 4 k < ε Drawbacks Only works for special polynomials. Among the inputs n ≤ x , this construction yields only O ( log x ) values F ( n ) that are n ε -friable. Friable values of polynomials Greg Martin

  10. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment From the last slide F ( n ) = 3 n 5 + 7 � Φ m ( − 3 4 7 ) F ( n k ) = − 7 n k = 3 8 k − 1 7 2 k m | ( 10 k − 1 ) � Φ m ( − 3 4 7 ) � � the primes dividing F ( n k ) are at most max � m | ( 10 k − 1 ) each Φ m ( x ) is roughly | x | φ ( m ) ≤ | x | φ ( 10 k − 1 ) n k is roughly ( 3 4 7 ) 4 k , but the largest prime factor of F ( n k ) is bounded by roughly ( 3 4 7 ) φ ( 10 k − 1 ) there are infinitely many k with φ ( 10 k − 1 ) / 4 k < ε Drawbacks Only works for special polynomials. Among the inputs n ≤ x , this construction yields only O ( log x ) values F ( n ) that are n ε -friable. Friable values of polynomials Greg Martin

  11. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment From the last slide F ( n ) = 3 n 5 + 7 � Φ m ( − 3 4 7 ) F ( n k ) = − 7 n k = 3 8 k − 1 7 2 k m | ( 10 k − 1 ) � Φ m ( − 3 4 7 ) � � the primes dividing F ( n k ) are at most max � m | ( 10 k − 1 ) each Φ m ( x ) is roughly | x | φ ( m ) ≤ | x | φ ( 10 k − 1 ) n k is roughly ( 3 4 7 ) 4 k , but the largest prime factor of F ( n k ) is bounded by roughly ( 3 4 7 ) φ ( 10 k − 1 ) there are infinitely many k with φ ( 10 k − 1 ) / 4 k < ε Drawbacks Only works for special polynomials. Among the inputs n ≤ x , this construction yields only O ( log x ) values F ( n ) that are n ε -friable. Friable values of polynomials Greg Martin

  12. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment From the last slide F ( n ) = 3 n 5 + 7 � Φ m ( − 3 4 7 ) F ( n k ) = − 7 n k = 3 8 k − 1 7 2 k m | ( 10 k − 1 ) � Φ m ( − 3 4 7 ) � � the primes dividing F ( n k ) are at most max � m | ( 10 k − 1 ) each Φ m ( x ) is roughly | x | φ ( m ) ≤ | x | φ ( 10 k − 1 ) n k is roughly ( 3 4 7 ) 4 k , but the largest prime factor of F ( n k ) is bounded by roughly ( 3 4 7 ) φ ( 10 k − 1 ) there are infinitely many k with φ ( 10 k − 1 ) / 4 k < ε Drawbacks Only works for special polynomials. Among the inputs n ≤ x , this construction yields only O ( log x ) values F ( n ) that are n ε -friable. Friable values of polynomials Greg Martin

  13. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment From the last slide F ( n ) = 3 n 5 + 7 � Φ m ( − 3 4 7 ) F ( n k ) = − 7 n k = 3 8 k − 1 7 2 k m | ( 10 k − 1 ) � Φ m ( − 3 4 7 ) � � the primes dividing F ( n k ) are at most max � m | ( 10 k − 1 ) each Φ m ( x ) is roughly | x | φ ( m ) ≤ | x | φ ( 10 k − 1 ) n k is roughly ( 3 4 7 ) 4 k , but the largest prime factor of F ( n k ) is bounded by roughly ( 3 4 7 ) φ ( 10 k − 1 ) there are infinitely many k with φ ( 10 k − 1 ) / 4 k < ε Drawbacks Only works for special polynomials. Among the inputs n ≤ x , this construction yields only O ( log x ) values F ( n ) that are n ε -friable. Friable values of polynomials Greg Martin

  14. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Polynomial factorizations Example The polynomial F ( x + F ( x )) is always divisible by F ( x ) . In particular, if deg F = d , then F ( x + F ( x )) is roughly x d 2 yet is automatically roughly x d 2 − d -friable. Mnemonic � � F x + F ( x ) ≡ F ( x ) ≡ 0 (mod F ( x ) ) Cool special case If F ( x ) is quadratic with leading coefficient a , then � x + 1 � F ( x + F ( x )) = F ( x ) · aF . a So if F ( x ) = x 2 + bx + c , then F ( x + F ( x )) = F ( x ) F ( x + 1 ) . Friable values of polynomials Greg Martin

  15. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Polynomial factorizations Example The polynomial F ( x + F ( x )) is always divisible by F ( x ) . In particular, if deg F = d , then F ( x + F ( x )) is roughly x d 2 yet is automatically roughly x d 2 − d -friable. Mnemonic � � F x + F ( x ) ≡ F ( x ) ≡ 0 (mod F ( x ) ) Cool special case If F ( x ) is quadratic with leading coefficient a , then � x + 1 � F ( x + F ( x )) = F ( x ) · aF . a So if F ( x ) = x 2 + bx + c , then F ( x + F ( x )) = F ( x ) F ( x + 1 ) . Friable values of polynomials Greg Martin

  16. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Polynomial factorizations Example The polynomial F ( x + F ( x )) is always divisible by F ( x ) . In particular, if deg F = d , then F ( x + F ( x )) is roughly x d 2 yet is automatically roughly x d 2 − d -friable. Mnemonic � � F x + F ( x ) ≡ F ( x ) ≡ 0 (mod F ( x ) ) Cool special case If F ( x ) is quadratic with leading coefficient a , then � x + 1 � F ( x + F ( x )) = F ( x ) · aF . a So if F ( x ) = x 2 + bx + c , then F ( x + F ( x )) = F ( x ) F ( x + 1 ) . Friable values of polynomials Greg Martin

  17. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment A refinement of Schinzel Idea: use the reciprocal polynomial x d F ( 1 / x ) . Proposition Let h ( x ) be a polynomial such that xh ( x ) − 1 is divisible by x d F ( 1 / x ) . Then F ( h ( x )) is divisible by x d F ( 1 / x ) . In particular, we can take deg h = d − 1 , in which case F ( h ( x )) is roughly x d 2 − d yet is automatically roughly x d 2 − 2 d -friable. Note: The proposition isn’t true for d = 2 , since the leftover “factor” of degree 2 2 − 2 · 2 = 0 is a constant. Mnemonic F ( h ( x )) ≡ F ( 1 / x ) ≡ 0 (mod F ( 1 / x ) ) Friable values of polynomials Greg Martin

  18. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment A refinement of Schinzel Idea: use the reciprocal polynomial x d F ( 1 / x ) . Proposition Let h ( x ) be a polynomial such that xh ( x ) − 1 is divisible by x d F ( 1 / x ) . Then F ( h ( x )) is divisible by x d F ( 1 / x ) . In particular, we can take deg h = d − 1 , in which case F ( h ( x )) is roughly x d 2 − d yet is automatically roughly x d 2 − 2 d -friable. Note: The proposition isn’t true for d = 2 , since the leftover “factor” of degree 2 2 − 2 · 2 = 0 is a constant. Mnemonic F ( h ( x )) ≡ F ( 1 / x ) ≡ 0 (mod F ( 1 / x ) ) Friable values of polynomials Greg Martin

  19. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment A refinement of Schinzel Idea: use the reciprocal polynomial x d F ( 1 / x ) . Proposition Let h ( x ) be a polynomial such that xh ( x ) − 1 is divisible by x d F ( 1 / x ) . Then F ( h ( x )) is divisible by x d F ( 1 / x ) . In particular, we can take deg h = d − 1 , in which case F ( h ( x )) is roughly x d 2 − d yet is automatically roughly x d 2 − 2 d -friable. Note: The proposition isn’t true for d = 2 , since the leftover “factor” of degree 2 2 − 2 · 2 = 0 is a constant. Mnemonic F ( h ( x )) ≡ F ( 1 / x ) ≡ 0 (mod F ( 1 / x ) ) Friable values of polynomials Greg Martin

  20. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment A refinement of Schinzel Idea: use the reciprocal polynomial x d F ( 1 / x ) . Proposition Let h ( x ) be a polynomial such that xh ( x ) − 1 is divisible by x d F ( 1 / x ) . Then F ( h ( x )) is divisible by x d F ( 1 / x ) . In particular, we can take deg h = d − 1 , in which case F ( h ( x )) is roughly x d 2 − d yet is automatically roughly x d 2 − 2 d -friable. Note: The proposition isn’t true for d = 2 , since the leftover “factor” of degree 2 2 − 2 · 2 = 0 is a constant. Mnemonic F ( h ( x )) ≡ F ( 1 / x ) ≡ 0 (mod F ( 1 / x ) ) Friable values of polynomials Greg Martin

  21. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Recursively use Schinzel’s construction Let D m denote an unspecified polynomial of degree m . Schinzel’s construction: if deg F = d , we can find D d − 1 such that F ( D d − 1 ) = D d D d ( d − 2 ) . Example: deg F ( x ) = 4 Use Schinzel’s construction repeatedly: D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 D 84 = F ( D 3 ( D 7 ) ) = D 28 D 8 D 48 “score” = 48/21 D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 2208/987 For deg F = 2 , begin with F ( D 4 ) = D 2 D 2 D 4 . Specifically, � � �� � x + 1 � x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . F a For deg F = 3 , begin with F ( D 4 ) = D 3 D 3 D 6 . Friable values of polynomials Greg Martin

  22. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Recursively use Schinzel’s construction Let D m denote an unspecified polynomial of degree m . Schinzel’s construction: if deg F = d , we can find D d − 1 such that F ( D d − 1 ) = D d D d ( d − 2 ) . Example: deg F ( x ) = 4 Use Schinzel’s construction repeatedly: D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 D 84 = F ( D 3 ( D 7 ) ) = D 28 D 8 D 48 “score” = 48/21 D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 2208/987 For deg F = 2 , begin with F ( D 4 ) = D 2 D 2 D 4 . Specifically, � � �� � x + 1 � x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . F a For deg F = 3 , begin with F ( D 4 ) = D 3 D 3 D 6 . Friable values of polynomials Greg Martin

  23. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Recursively use Schinzel’s construction Let D m denote an unspecified polynomial of degree m . Schinzel’s construction: if deg F = d , we can find D d − 1 such that F ( D d − 1 ) = D d D d ( d − 2 ) . Example: deg F ( x ) = 4 Use Schinzel’s construction repeatedly: D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 D 84 = F ( D 3 ( D 7 ) ) = D 28 D 8 D 48 “score” = 48/21 D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 2208/987 For deg F = 2 , begin with F ( D 4 ) = D 2 D 2 D 4 . Specifically, � � �� � x + 1 � x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . F a For deg F = 3 , begin with F ( D 4 ) = D 3 D 3 D 6 . Friable values of polynomials Greg Martin

  24. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Recursively use Schinzel’s construction Let D m denote an unspecified polynomial of degree m . Schinzel’s construction: if deg F = d , we can find D d − 1 such that F ( D d − 1 ) = D d D d ( d − 2 ) . Example: deg F ( x ) = 4 Use Schinzel’s construction repeatedly: D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 D 84 = F ( D 3 ( D 7 ) ) = D 28 D 8 D 48 “score” = 48/21 D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 2208/987 For deg F = 2 , begin with F ( D 4 ) = D 2 D 2 D 4 . Specifically, � � �� � x + 1 � x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . F a For deg F = 3 , begin with F ( D 4 ) = D 3 D 3 D 6 . Friable values of polynomials Greg Martin

  25. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Recursively use Schinzel’s construction Let D m denote an unspecified polynomial of degree m . Schinzel’s construction: if deg F = d , we can find D d − 1 such that F ( D d − 1 ) = D d D d ( d − 2 ) . Example: deg F ( x ) = 4 Use Schinzel’s construction repeatedly: D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 D 84 = F ( ) = D 28 D 8 D 48 “score” = 48/21 D 21 D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 2208/987 For deg F = 2 , begin with F ( D 4 ) = D 2 D 2 D 4 . Specifically, � � �� � x + 1 � x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . F a For deg F = 3 , begin with F ( D 4 ) = D 3 D 3 D 6 . Friable values of polynomials Greg Martin

  26. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Recursively use Schinzel’s construction Let D m denote an unspecified polynomial of degree m . Schinzel’s construction: if deg F = d , we can find D d − 1 such that F ( D d − 1 ) = D d D d ( d − 2 ) . Example: deg F ( x ) = 4 Use Schinzel’s construction repeatedly: D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 D 84 = F ( ) = D 28 D 8 D 48 “score” = 48/21 D 21 D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 2208/987 For deg F = 2 , begin with F ( D 4 ) = D 2 D 2 D 4 . Specifically, � � �� � x + 1 � x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . F a For deg F = 3 , begin with F ( D 4 ) = D 3 D 3 D 6 . Friable values of polynomials Greg Martin

  27. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Recursively use Schinzel’s construction Let D m denote an unspecified polynomial of degree m . Schinzel’s construction: if deg F = d , we can find D d − 1 such that F ( D d − 1 ) = D d D d ( d − 2 ) . Example: deg F ( x ) = 4 Use Schinzel’s construction repeatedly: D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 D 84 = F ( ) = D 28 D 8 D 48 “score” = 48/21 D 21 D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 2208/987 For deg F = 2 , begin with F ( D 4 ) = D 2 D 2 D 4 . Specifically, � � �� � x + 1 � x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . F a For deg F = 3 , begin with F ( D 4 ) = D 3 D 3 D 6 . Friable values of polynomials Greg Martin

  28. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Recursively use Schinzel’s construction Let D m denote an unspecified polynomial of degree m . Schinzel’s construction: if deg F = d , we can find D d − 1 such that F ( D d − 1 ) = D d D d ( d − 2 ) . Example: deg F ( x ) = 4 Use Schinzel’s construction repeatedly: D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 D 84 = F ( ) = D 28 D 8 D 48 “score” = 48/21 D 21 D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 2208/987 For deg F = 2 , begin with F ( D 4 ) = D 2 D 2 D 4 . Specifically, � � �� � x + 1 � x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . F a For deg F = 3 , begin with F ( D 4 ) = D 3 D 3 D 6 . Friable values of polynomials Greg Martin

  29. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Recursively use Schinzel’s construction Let D m denote an unspecified polynomial of degree m . Schinzel’s construction: if deg F = d , we can find D d − 1 such that F ( D d − 1 ) = D d D d ( d − 2 ) . Example: deg F ( x ) = 4 Use Schinzel’s construction repeatedly: D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 D 84 = F ( ) = D 28 D 8 D 48 “score” = 48/21 D 21 D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 2208/987 For deg F = 2 , begin with F ( D 4 ) = D 2 D 2 D 4 . Specifically, � � �� � x + 1 � x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . F a For deg F = 3 , begin with F ( D 4 ) = D 3 D 3 D 6 . Friable values of polynomials Greg Martin

  30. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment How friable can values of general polynomials be? ∞ � � 1 � For d ≥ 4 , define s ( d ) = d 1 − , where u j ( d ) j = 1 u 1 ( d ) = d − 1 and u j + 1 ( d ) = u j ( d ) 2 − 2 Define s ( 2 ) = s ( 4 ) / 4 and s ( 3 ) = s ( 6 ) / 4 Theorem (Schinzel, 1967) Given a polynomial F ( x ) of degree d ≥ 2 , there are infinitely many numbers n for which F ( n ) is n s ( d ) -friable. F ( n ) s ( d ) F ( n ) s ( d ) ε degree 1 degree 5 3.46410 degree 2 0.55902 degree 6 4.58258 degree 3 1.14564 degree 7 5.65685 degree 4 2.23607 degree d ≈ d − 1 − 2 / d Friable values of polynomials Greg Martin

  31. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment How friable can values of general polynomials be? ∞ � � 1 � For d ≥ 4 , define s ( d ) = d 1 − , where u j ( d ) j = 1 u 1 ( d ) = d − 1 and u j + 1 ( d ) = u j ( d ) 2 − 2 Define s ( 2 ) = s ( 4 ) / 4 and s ( 3 ) = s ( 6 ) / 4 Theorem (Schinzel, 1967) Given a polynomial F ( x ) of degree d ≥ 2 , there are infinitely many numbers n for which F ( n ) is n s ( d ) -friable. F ( n ) s ( d ) F ( n ) s ( d ) ε degree 1 degree 5 3.46410 degree 2 0.55902 degree 6 4.58258 degree 3 1.14564 degree 7 5.65685 degree 4 2.23607 degree d ≈ d − 1 − 2 / d Friable values of polynomials Greg Martin

  32. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment How friable can values of general polynomials be? ∞ � � 1 � For d ≥ 4 , define s ( d ) = d 1 − , where u j ( d ) j = 1 u 1 ( d ) = d − 1 and u j + 1 ( d ) = u j ( d ) 2 − 2 Define s ( 2 ) = s ( 4 ) / 4 and s ( 3 ) = s ( 6 ) / 4 Theorem (Schinzel, 1967) Given a polynomial F ( x ) of degree d ≥ 2 , there are infinitely many numbers n for which F ( n ) is n s ( d ) -friable. F ( n ) s ( d ) F ( n ) s ( d ) ε degree 1 degree 5 3.46410 degree 2 0.55902 degree 6 4.58258 degree 3 1.14564 degree 7 5.65685 degree 4 2.23607 degree d ≈ d − 1 − 2 / d Friable values of polynomials Greg Martin

  33. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment How friable can values of general polynomials be? ∞ � � 1 � For d ≥ 4 , define s ( d ) = d 1 − , where u j ( d ) j = 1 u 1 ( d ) = d − 1 and u j + 1 ( d ) = u j ( d ) 2 − 2 Define s ( 2 ) = s ( 4 ) / 4 and s ( 3 ) = s ( 6 ) / 4 Theorem (Schinzel, 1967) Given a polynomial F ( x ) of degree d ≥ 2 , there are infinitely many numbers n for which F ( n ) is n s ( d ) -friable. F ( n ) s ( d ) F ( n ) s ( d ) ε degree 1 degree 5 3.46410 degree 2 0.55902 degree 6 4.58258 degree 3 1.14564 degree 7 5.65685 degree 4 2.23607 degree d ≈ d − 1 − 2 / d Friable values of polynomials Greg Martin

  34. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Polynomial substitution yields few friable values Special case Given a quadratic polynomial F ( x ) , there are infinitely many numbers n for which F ( n ) is n 0 . 55902 -friable. Example To obtain n for which F ( n ) is n 0 . 56 -friable: D 168 = F ( D 84 ) = D 42 D 42 D 28 D 8 D 48 “score” = 48 / 84 > 0 . 56 D 7896 = F ( D 3948 ) “score” = 2208 / 3948 = D 1974 D 1974 D 1316 D 376 D 48 D 2208 < 0 . 56 The counting function of such n is about x 1 / 3948 . “Improvement” Balog, M., Wooley (unpublished) can obtain x 2 / 3948 and an analogous improvement for deg F = 3 . Friable values of polynomials Greg Martin

  35. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Polynomial substitution yields few friable values Special case Given a quadratic polynomial F ( x ) , there are infinitely many numbers n for which F ( n ) is n 0 . 55902 -friable. Example To obtain n for which F ( n ) is n 0 . 56 -friable: D 168 = F ( D 84 ) = D 42 D 42 D 28 D 8 D 48 “score” = 48 / 84 > 0 . 56 D 7896 = F ( D 3948 ) “score” = 2208 / 3948 = D 1974 D 1974 D 1316 D 376 D 48 D 2208 < 0 . 56 The counting function of such n is about x 1 / 3948 . “Improvement” Balog, M., Wooley (unpublished) can obtain x 2 / 3948 and an analogous improvement for deg F = 3 . Friable values of polynomials Greg Martin

  36. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Polynomial substitution yields few friable values Special case Given a quadratic polynomial F ( x ) , there are infinitely many numbers n for which F ( n ) is n 0 . 55902 -friable. Example To obtain n for which F ( n ) is n 0 . 56 -friable: D 168 = F ( D 84 ) = D 42 D 42 D 28 D 8 D 48 “score” = 48 / 84 > 0 . 56 D 7896 = F ( D 3948 ) “score” = 2208 / 3948 = D 1974 D 1974 D 1316 D 376 D 48 D 2208 < 0 . 56 The counting function of such n is about x 1 / 3948 . “Improvement” Balog, M., Wooley (unpublished) can obtain x 2 / 3948 and an analogous improvement for deg F = 3 . Friable values of polynomials Greg Martin

  37. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Polynomial substitution yields few friable values Special case Given a quadratic polynomial F ( x ) , there are infinitely many numbers n for which F ( n ) is n 0 . 55902 -friable. Example To obtain n for which F ( n ) is n 0 . 56 -friable: D 168 = F ( D 84 ) = D 42 D 42 D 28 D 8 D 48 “score” = 48 / 84 > 0 . 56 D 7896 = F ( D 3948 ) “score” = 2208 / 3948 = D 1974 D 1974 D 1316 D 376 D 48 D 2208 < 0 . 56 The counting function of such n is about x 1 / 3948 . “Improvement” Balog, M., Wooley (unpublished) can obtain x 2 / 3948 and an analogous improvement for deg F = 3 . Friable values of polynomials Greg Martin

  38. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Polynomial substitution yields few friable values Special case Given a quadratic polynomial F ( x ) , there are infinitely many numbers n for which F ( n ) is n 0 . 55902 -friable. Example To obtain n for which F ( n ) is n 0 . 56 -friable: D 168 = F ( D 84 ) = D 42 D 42 D 28 D 8 D 48 “score” = 48 / 84 > 0 . 56 D 7896 = F ( D 3948 ) “score” = 2208 / 3948 = D 1974 D 1974 D 1316 D 376 D 48 D 2208 < 0 . 56 The counting function of such n is about x 1 / 3948 . “Improvement” Balog, M., Wooley (unpublished) can obtain x 2 / 3948 and an analogous improvement for deg F = 3 . Friable values of polynomials Greg Martin

  39. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Can we have lots of friable values? Our expectation For any ε > 0 , a positive proportion of values F ( n ) are n ε -friable. At the turn of the millenium, we knew this for: linear polynomials (arithmetic progressions) Hildebrand (1985), then Balog and Ruzsa (1995): F ( n ) = n ( an + b ) Hildebrand (1989): F ( n ) = ( n + 1 ) · · · ( n + L ) , positive proportion of values n β -friable for β > e − 1 / ( L − 1 ) L , so β ≤ e − 1 / L is nontrivial Note: ρ ( e − 1 / L ) = 1 − 1 Dartyge (1996): F ( n ) = n 2 + 1 , positive proportion of values n β -friable for β > 149 / 179 Friable values of polynomials Greg Martin

  40. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Can we have lots of friable values? Our expectation For any ε > 0 , a positive proportion of values F ( n ) are n ε -friable. At the turn of the millenium, we knew this for: linear polynomials (arithmetic progressions) Hildebrand (1985), then Balog and Ruzsa (1995): F ( n ) = n ( an + b ) Hildebrand (1989): F ( n ) = ( n + 1 ) · · · ( n + L ) , positive proportion of values n β -friable for β > e − 1 / ( L − 1 ) L , so β ≤ e − 1 / L is nontrivial Note: ρ ( e − 1 / L ) = 1 − 1 Dartyge (1996): F ( n ) = n 2 + 1 , positive proportion of values n β -friable for β > 149 / 179 Friable values of polynomials Greg Martin

  41. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Can we have lots of friable values? Our expectation For any ε > 0 , a positive proportion of values F ( n ) are n ε -friable. At the turn of the millenium, we knew this for: linear polynomials (arithmetic progressions) Hildebrand (1985), then Balog and Ruzsa (1995): F ( n ) = n ( an + b ) Hildebrand (1989): F ( n ) = ( n + 1 ) · · · ( n + L ) , positive proportion of values n β -friable for β > e − 1 / ( L − 1 ) L , so β ≤ e − 1 / L is nontrivial Note: ρ ( e − 1 / L ) = 1 − 1 Dartyge (1996): F ( n ) = n 2 + 1 , positive proportion of values n β -friable for β > 149 / 179 Friable values of polynomials Greg Martin

  42. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Can we have lots of friable values? Our expectation For any ε > 0 , a positive proportion of values F ( n ) are n ε -friable. At the turn of the millenium, we knew this for: linear polynomials (arithmetic progressions) Hildebrand (1985), then Balog and Ruzsa (1995): F ( n ) = n ( an + b ) Hildebrand (1989): F ( n ) = ( n + 1 ) · · · ( n + L ) , positive proportion of values n β -friable for β > e − 1 / ( L − 1 ) L , so β ≤ e − 1 / L is nontrivial Note: ρ ( e − 1 / L ) = 1 − 1 Dartyge (1996): F ( n ) = n 2 + 1 , positive proportion of values n β -friable for β > 149 / 179 Friable values of polynomials Greg Martin

  43. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Can we have lots of friable values? Our expectation For any ε > 0 , a positive proportion of values F ( n ) are n ε -friable. At the turn of the millenium, we knew this for: linear polynomials (arithmetic progressions) Hildebrand (1985), then Balog and Ruzsa (1995): F ( n ) = n ( an + b ) Hildebrand (1989): F ( n ) = ( n + 1 ) · · · ( n + L ) , positive proportion of values n β -friable for β > e − 1 / ( L − 1 ) L , so β ≤ e − 1 / L is nontrivial Note: ρ ( e − 1 / L ) = 1 − 1 Dartyge (1996): F ( n ) = n 2 + 1 , positive proportion of values n β -friable for β > 149 / 179 Friable values of polynomials Greg Martin

  44. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Theorem (Dartyge, M., Tenenbaum, 2001) Let F ( x ) be the product of K distinct irreducible polynomials of degree d . Then for any ε > 0 , a positive proportion of values F ( n ) are n d − 1 / K + ε -friable. Anything better than n d -friable is nontrivial. No loss of generality When the friability level exceeds n d − 1 , only irreducible factors of degree at least d matter. Therefore the theorem also holds if F ( x ) has K distinct irreducible factors of degree d and any number of irreducible factors of degree less than d . Friable values of polynomials Greg Martin

  45. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Theorem (Dartyge, M., Tenenbaum, 2001) Let F ( x ) be the product of K distinct irreducible polynomials of degree d . Then for any ε > 0 , a positive proportion of values F ( n ) are n d − 1 / K + ε -friable. Anything better than n d -friable is nontrivial. No loss of generality When the friability level exceeds n d − 1 , only irreducible factors of degree at least d matter. Therefore the theorem also holds if F ( x ) has K distinct irreducible factors of degree d and any number of irreducible factors of degree less than d . Friable values of polynomials Greg Martin

  46. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Theorem (Dartyge, M., Tenenbaum, 2001) Let F ( x ) be the product of K distinct irreducible polynomials of degree d . Then for any ε > 0 , a positive proportion of values F ( n ) are n d − 1 / K + ε -friable. Anything better than n d -friable is nontrivial. No loss of generality When the friability level exceeds n d − 1 , only irreducible factors of degree at least d matter. Therefore the theorem also holds if F ( x ) has K distinct irreducible factors of degree d and any number of irreducible factors of degree less than d . Friable values of polynomials Greg Martin

  47. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) Definition π ( F ; x ) = # { n ≤ x : f ( n ) is prime for each irreducible factor f of F } Conjecture: π ( F ; x ) is asymptotic to H ( F ) li ( F ; x ) , where: � � � 1 x � li ( F ; x ) = dt ∼ log | f ( t ) | ( log x ) L f | F 0 < t < x f irreducible L : the number of distinct irreducible factors of F � − L � � � 1 − 1 1 − σ ( F ; p ) � H ( F ) = p p p σ ( F ; n ) : the number of solutions of F ( a ) ≡ 0 (mod n ) Friable values of polynomials Greg Martin

  48. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) Definition π ( F ; x ) = # { n ≤ x : f ( n ) is prime for each irreducible factor f of F } Conjecture: π ( F ; x ) is asymptotic to H ( F ) li ( F ; x ) , where: � � � 1 x � li ( F ; x ) = dt ∼ log | f ( t ) | ( log x ) L f | F 0 < t < x f irreducible L : the number of distinct irreducible factors of F � − L � � � 1 − 1 1 − σ ( F ; p ) � H ( F ) = p p p σ ( F ; n ) : the number of solutions of F ( a ) ≡ 0 (mod n ) Friable values of polynomials Greg Martin

  49. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) Definition π ( F ; x ) = # { n ≤ x : f ( n ) is prime for each irreducible factor f of F } Conjecture: π ( F ; x ) is asymptotic to H ( F ) li ( F ; x ) , where: � � � 1 x � li ( F ; x ) = dt ∼ log | f ( t ) | ( log x ) L f | F 0 < t < x f irreducible L : the number of distinct irreducible factors of F � − L � � � 1 − 1 1 − σ ( F ; p ) � H ( F ) = p p p σ ( F ; n ) : the number of solutions of F ( a ) ≡ 0 (mod n ) Friable values of polynomials Greg Martin

  50. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) Definition π ( F ; x ) = # { n ≤ x : f ( n ) is prime for each irreducible factor f of F } Conjecture: π ( F ; x ) is asymptotic to H ( F ) li ( F ; x ) , where: � � � 1 x � li ( F ; x ) = dt ∼ log | f ( t ) | ( log x ) L f | F 0 < t < x f irreducible L : the number of distinct irreducible factors of F � − L � � � 1 − 1 1 − σ ( F ; p ) � H ( F ) = p p p σ ( F ; n ) : the number of solutions of F ( a ) ≡ 0 (mod n ) Friable values of polynomials Greg Martin

  51. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) Definition π ( F ; x ) = # { n ≤ x : f ( n ) is prime for each irreducible factor f of F } Conjecture: π ( F ; x ) is asymptotic to H ( F ) li ( F ; x ) , where: � � � 1 x � li ( F ; x ) = dt ∼ log | f ( t ) | ( log x ) L f | F 0 < t < x f irreducible L : the number of distinct irreducible factors of F � − L � � � 1 − 1 1 − σ ( F ; p ) � H ( F ) = p p p σ ( F ; n ) : the number of solutions of F ( a ) ≡ 0 (mod n ) Friable values of polynomials Greg Martin

  52. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment What’s the connection? Why are we talking about prime values of polynomials in a lecture about friable values of polynomials? Heuristic The process of “guessing” the right answer for π ( F ; x ) puts us in the right state of mind for trying to guess the right answer for Ψ( F ; x , y ) . Implication By assuming the expected formula for π ( F ; x ) with some uniformity, we can derive a formula for Ψ( F ; x , y ) in a limited range. This implication informs our beliefs about the expected formula for Ψ( F ; x , y ) . Friable values of polynomials Greg Martin

  53. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment What’s the connection? Why are we talking about prime values of polynomials in a lecture about friable values of polynomials? Heuristic The process of “guessing” the right answer for π ( F ; x ) puts us in the right state of mind for trying to guess the right answer for Ψ( F ; x , y ) . Implication By assuming the expected formula for π ( F ; x ) with some uniformity, we can derive a formula for Ψ( F ; x , y ) in a limited range. This implication informs our beliefs about the expected formula for Ψ( F ; x , y ) . Friable values of polynomials Greg Martin

  54. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment What’s the connection? Why are we talking about prime values of polynomials in a lecture about friable values of polynomials? Heuristic The process of “guessing” the right answer for π ( F ; x ) puts us in the right state of mind for trying to guess the right answer for Ψ( F ; x , y ) . Implication By assuming the expected formula for π ( F ; x ) with some uniformity, we can derive a formula for Ψ( F ; x , y ) in a limited range. This implication informs our beliefs about the expected formula for Ψ( F ; x , y ) . Friable values of polynomials Greg Martin

  55. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment What’s the connection? Why are we talking about prime values of polynomials in a lecture about friable values of polynomials? Heuristic The process of “guessing” the right answer for π ( F ; x ) puts us in the right state of mind for trying to guess the right answer for Ψ( F ; x , y ) . Implication By assuming the expected formula for π ( F ; x ) with some uniformity, we can derive a formula for Ψ( F ; x , y ) in a limited range. This implication informs our beliefs about the expected formula for Ψ( F ; x , y ) . Friable values of polynomials Greg Martin

  56. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment What would we expect on probablistic grounds? Let F ( x ) = f 1 ( x ) · · · f L ( x ) , where deg f j ( x ) = d j . Let u > 0 . f j ( n ) is roughly n d j , and integers of that size are n 1 / u -friable with probability ρ ( d j u ) . Are the friabilities of the various factors f j ( n ) independent? This would lead to a prediction involving L � x ρ ( d j u ) . j = 1 What about local densities depending on the arithmetic of F (as in Hypothesis H)? Friable values of polynomials Greg Martin

  57. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment What would we expect on probablistic grounds? Let F ( x ) = f 1 ( x ) · · · f L ( x ) , where deg f j ( x ) = d j . Let u > 0 . f j ( n ) is roughly n d j , and integers of that size are n 1 / u -friable with probability ρ ( d j u ) . Are the friabilities of the various factors f j ( n ) independent? This would lead to a prediction involving L � x ρ ( d j u ) . j = 1 What about local densities depending on the arithmetic of F (as in Hypothesis H)? Friable values of polynomials Greg Martin

  58. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment What would we expect on probablistic grounds? Let F ( x ) = f 1 ( x ) · · · f L ( x ) , where deg f j ( x ) = d j . Let u > 0 . f j ( n ) is roughly n d j , and integers of that size are n 1 / u -friable with probability ρ ( d j u ) . Are the friabilities of the various factors f j ( n ) independent? This would lead to a prediction involving L � x ρ ( d j u ) . j = 1 What about local densities depending on the arithmetic of F (as in Hypothesis H)? Friable values of polynomials Greg Martin

  59. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment What would we expect on probablistic grounds? Let F ( x ) = f 1 ( x ) · · · f L ( x ) , where deg f j ( x ) = d j . Let u > 0 . f j ( n ) is roughly n d j , and integers of that size are n 1 / u -friable with probability ρ ( d j u ) . Are the friabilities of the various factors f j ( n ) independent? This would lead to a prediction involving L � x ρ ( d j u ) . j = 1 What about local densities depending on the arithmetic of F (as in Hypothesis H)? Friable values of polynomials Greg Martin

  60. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Conjecture for friable values of polynomials Conjecture Let F ( x ) be any polynomial, let f 1 , . . . , f L be its distinct irreducible factors, and let d 1 , . . . , d L be their degrees. Then L � � x Ψ( F ; x , x 1 / u ) = x � ρ ( d j u ) + O log x j = 1 for all 0 < u . If F irreducible: Ψ( F ; x , x 1 / u ) = x ρ ( du ) + O ( x / log x ) for 0 < u . Remark: Not as universally accepted as Hypothesis H; lack of local factors is controversial. Friable values of polynomials Greg Martin

  61. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Conjecture for friable values of polynomials Conjecture Let F ( x ) be any polynomial, let f 1 , . . . , f L be its distinct irreducible factors, and let d 1 , . . . , d L be their degrees. Then L � � x Ψ( F ; x , x 1 / u ) = x � ρ ( d j u ) + O log x j = 1 for all 0 < u . If F irreducible: Ψ( F ; x , x 1 / u ) = x ρ ( du ) + O ( x / log x ) for 0 < u . Remark: Not as universally accepted as Hypothesis H; lack of local factors is controversial. Friable values of polynomials Greg Martin

  62. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Conjecture for friable values of polynomials Conjecture Let F ( x ) be any polynomial, let f 1 , . . . , f L be its distinct irreducible factors, and let d 1 , . . . , d L be their degrees. Then L � � x Ψ( F ; x , x 1 / u ) = x � ρ ( d j u ) + O log x j = 1 for all 0 < u . If F irreducible: Ψ( F ; x , x 1 / u ) = x ρ ( du ) + O ( x / log x ) for 0 < u . Remark: Not as universally accepted as Hypothesis H; lack of local factors is controversial. Friable values of polynomials Greg Martin

  63. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Conjecture for friable values of polynomials Theorem (M., 2002) Assume (a uniform version of) Hypothesis UH. Let F ( x ) be any polynomial, let f 1 , . . . , f L be its distinct irreducible factors, and let d 1 , . . . , d L be their degrees. Let d = max { d 1 , . . . , d L } , and let F have exactly K distinct irreducible factors of degree d . Then L � � x Ψ( F ; x , x 1 / u ) = x � ρ ( d j u ) + O log x j = 1 for all 0 < u < 1 / ( d − 1 / K ) . If F irreducible: Ψ( F ; x , x 1 / u ) = x ρ ( du ) + O ( x / log x ) for 0 < u < 1 / ( d − 1 ) . Trivial: 0 < u < 1 / d . Friable values of polynomials Greg Martin

  64. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Reduction to (analytically) convenient polynomials Without loss of generality, we may assume: F ( x ) is the product of distinct irreducible polynomials f 1 ( x ) , . . . , f K ( x ) , all of the same degree d . Acceptable since the friability level y exceeds x d − 1 , so factors of degree d − 1 or less are automatically y -friable. In this situation, what we want to prove is that � � x Ψ( F ; x , x 1 / u ) = x ρ ( du ) K + O log x for all 0 < u < 1 / ( d − 1 / K ) . Friable values of polynomials Greg Martin

  65. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Reduction to (analytically) convenient polynomials Without loss of generality, we may assume: F ( x ) is the product of distinct irreducible polynomials f 1 ( x ) , . . . , f K ( x ) , all of the same degree d . Acceptable since the friability level y exceeds x d − 1 , so factors of degree d − 1 or less are automatically y -friable. In this situation, what we want to prove is that � � x Ψ( F ; x , x 1 / u ) = x ρ ( du ) K + O log x for all 0 < u < 1 / ( d − 1 / K ) . Friable values of polynomials Greg Martin

  66. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Reduction to (analytically) convenient polynomials Without loss of generality, we may assume: F ( x ) is the product of distinct irreducible polynomials f 1 ( x ) , . . . , f K ( x ) , all of the same degree d . Acceptable since the friability level y exceeds x d − 1 , so factors of degree d − 1 or less are automatically y -friable. In this situation, what we want to prove is that � � x Ψ( F ; x , x 1 / u ) = x ρ ( du ) K + O log x for all 0 < u < 1 / ( d − 1 / K ) . Friable values of polynomials Greg Martin

  67. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Reduction to (algebraically) convenient polynomials Without loss of generality, we may assume: F ( x ) takes at least one nonzero value modulo every prime. 1 No two distinct irreducible factors f i ( x ) , f j ( x ) of F ( x ) have a 2 common zero modulo any prime. We’ll call such a polynomial a nice polynomial. (1) is not a necessary condition to have friable values (as it is to have prime values). Nevertheless, we can still reduce to this case. Both (1) and (2) can be achieved by looking at values of F ( x ) on suitable arithmetic progressions F ( Qx + R ) separately. Friable values of polynomials Greg Martin

  68. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Reduction to (algebraically) convenient polynomials Without loss of generality, we may assume: F ( x ) takes at least one nonzero value modulo every prime. 1 No two distinct irreducible factors f i ( x ) , f j ( x ) of F ( x ) have a 2 common zero modulo any prime. We’ll call such a polynomial a nice polynomial. (1) is not a necessary condition to have friable values (as it is to have prime values). Nevertheless, we can still reduce to this case. Both (1) and (2) can be achieved by looking at values of F ( x ) on suitable arithmetic progressions F ( Qx + R ) separately. Friable values of polynomials Greg Martin

  69. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Reduction to (algebraically) convenient polynomials Without loss of generality, we may assume: F ( x ) takes at least one nonzero value modulo every prime. 1 No two distinct irreducible factors f i ( x ) , f j ( x ) of F ( x ) have a 2 common zero modulo any prime. We’ll call such a polynomial a nice polynomial. (1) is not a necessary condition to have friable values (as it is to have prime values). Nevertheless, we can still reduce to this case. Both (1) and (2) can be achieved by looking at values of F ( x ) on suitable arithmetic progressions F ( Qx + R ) separately. Friable values of polynomials Greg Martin

  70. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Reduction to (algebraically) convenient polynomials Without loss of generality, we may assume: F ( x ) takes at least one nonzero value modulo every prime. 1 No two distinct irreducible factors f i ( x ) , f j ( x ) of F ( x ) have a 2 common zero modulo any prime. We’ll call such a polynomial a nice polynomial. (1) is not a necessary condition to have friable values (as it is to have prime values). Nevertheless, we can still reduce to this case. Both (1) and (2) can be achieved by looking at values of F ( x ) on suitable arithmetic progressions F ( Qx + R ) separately. Friable values of polynomials Greg Martin

  71. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Reduction to (algebraically) convenient polynomials Without loss of generality, we may assume: F ( x ) takes at least one nonzero value modulo every prime. 1 No two distinct irreducible factors f i ( x ) , f j ( x ) of F ( x ) have a 2 common zero modulo any prime. We’ll call such a polynomial a nice polynomial. (1) is not a necessary condition to have friable values (as it is to have prime values). Nevertheless, we can still reduce to this case. Both (1) and (2) can be achieved by looking at values of F ( x ) on suitable arithmetic progressions F ( Qx + R ) separately. Friable values of polynomials Greg Martin

  72. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Inclusion-exclusion on irreducible factors Proposition Let F be a nice polynomial, and let F 1 , . . . , F K denote the distinct irreducible factors of F . Then for x ≥ y ≥ 1 , � � ( − 1 ) k Ψ( F ; x , y ) = ⌊ x ⌋ + M ( F i 1 . . . F i k ; x , y ) . 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K Even though we don’t know its definition yet: if we knew that M ( F i 1 . . . F i k ; x , x 1 / u ) ∼ x ( log du ) k , then Ψ( F ; x , x 1 / u ) ∼ x + � � ( − 1 ) k x ( log du ) k 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K � � � � K ( − log du ) k � = x 1 + k 1 ≤ k ≤ K = x ( 1 − log du ) K = x ρ ( du ) K . Friable values of polynomials Greg Martin

  73. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Inclusion-exclusion on irreducible factors Proposition Let F be a nice polynomial, and let F 1 , . . . , F K denote the distinct irreducible factors of F . Then for x ≥ y ≥ 1 , � � ( − 1 ) k Ψ( F ; x , y ) = ⌊ x ⌋ + M ( F i 1 . . . F i k ; x , y ) . 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K Even though we don’t know its definition yet: if we knew that M ( F i 1 . . . F i k ; x , x 1 / u ) ∼ x ( log du ) k , then Ψ( F ; x , x 1 / u ) ∼ x + � � ( − 1 ) k x ( log du ) k 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K � � � � K ( − log du ) k � = x 1 + k 1 ≤ k ≤ K = x ( 1 − log du ) K = x ρ ( du ) K . Friable values of polynomials Greg Martin

  74. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Inclusion-exclusion on irreducible factors Proposition Let F be a nice polynomial, and let F 1 , . . . , F K denote the distinct irreducible factors of F . Then for x ≥ y ≥ 1 , � � ( − 1 ) k Ψ( F ; x , y ) = ⌊ x ⌋ + M ( F i 1 . . . F i k ; x , y ) . 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K Even though we don’t know its definition yet: if we knew that M ( F i 1 . . . F i k ; x , x 1 / u ) ∼ x ( log du ) k , then Ψ( F ; x , x 1 / u ) ∼ x + � � ( − 1 ) k x ( log du ) k 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K � � � � K ( − log du ) k � = x 1 + k 1 ≤ k ≤ K = x ( 1 − log du ) K = x ρ ( du ) K . Friable values of polynomials Greg Martin

  75. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Inclusion-exclusion on irreducible factors Proposition Let F be a nice polynomial, and let F 1 , . . . , F K denote the distinct irreducible factors of F . Then for x ≥ y ≥ 1 , � � ( − 1 ) k Ψ( F ; x , y ) = ⌊ x ⌋ + M ( F i 1 . . . F i k ; x , y ) . 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K Even though we don’t know its definition yet: if we knew that M ( F i 1 . . . F i k ; x , x 1 / u ) ∼ x ( log du ) k , then Ψ( F ; x , x 1 / u ) ∼ x + � � ( − 1 ) k x ( log du ) k 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K � � � � K ( − log du ) k � = x 1 + k 1 ≤ k ≤ K = x ( 1 − log du ) K = x ρ ( du ) K . Friable values of polynomials Greg Martin

  76. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment The resulting counting problem Definition M ( f ; x , y ) = # { 1 ≤ n ≤ x : for each irreducible factor g of f , there exists a prime p > y such that p | g ( n ) } . We want to prove: M ( F i 1 . . . F i k ; x , x 1 / u ) ∼ x ( log du ) k Strategy: Sort by the values n j = F i j ( n ) / p j , among those n counted by M ( F i 1 . . . F i k ; x , x 1 / u ) (so p j is the prime > y corresponding to n ). So given n j , we focus on residue classes for n in which n j | F i j ( n ) ; we want to count how often F i j ( n ) / n j is prime. Friable values of polynomials Greg Martin

  77. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment The resulting counting problem Definition M ( f ; x , y ) = # { 1 ≤ n ≤ x : for each irreducible factor g of f , there exists a prime p > y such that p | g ( n ) } . We want to prove: M ( F i 1 . . . F i k ; x , x 1 / u ) ∼ x ( log du ) k Strategy: Sort by the values n j = F i j ( n ) / p j , among those n counted by M ( F i 1 . . . F i k ; x , x 1 / u ) (so p j is the prime > y corresponding to n ). So given n j , we focus on residue classes for n in which n j | F i j ( n ) ; we want to count how often F i j ( n ) / n j is prime. Friable values of polynomials Greg Martin

  78. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment The resulting counting problem Definition M ( f ; x , y ) = # { 1 ≤ n ≤ x : for each irreducible factor g of f , there exists a prime p > y such that p | g ( n ) } . We want to prove: M ( F i 1 . . . F i k ; x , x 1 / u ) ∼ x ( log du ) k Strategy: Sort by the values n j = F i j ( n ) / p j , among those n counted by M ( F i 1 . . . F i k ; x , x 1 / u ) (so p j is the prime > y corresponding to n ). So given n j , we focus on residue classes for n in which n j | F i j ( n ) ; we want to count how often F i j ( n ) / n j is prime. Friable values of polynomials Greg Martin

  79. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment The resulting counting problem Definition M ( f ; x , y ) = # { 1 ≤ n ≤ x : for each irreducible factor g of f , there exists a prime p > y such that p | g ( n ) } . We want to prove: M ( F i 1 . . . F i k ; x , x 1 / u ) ∼ x ( log du ) k Strategy: Sort by the values n j = F i j ( n ) / p j , among those n counted by M ( F i 1 . . . F i k ; x , x 1 / u ) (so p j is the prime > y corresponding to n ). So given n j , we focus on residue classes for n in which n j | F i j ( n ) ; we want to count how often F i j ( n ) / n j is prime. Friable values of polynomials Greg Martin

  80. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Recall notation: prime values of polynomials π ( F ; t ) = # { 1 ≤ n ≤ t : f ( n ) is prime for every irreducible factor f of F } Proposition For f = f 1 . . . f k and x and y sufficiently large, x − b � � � � � M ( f ; x , y ) ∼ · · · π f n 1 ··· n k , b ; . n 1 · · · n k n 1 ≤ x d / y n k ≤ x d / y b ∈R ( f ; n 1 ,..., n k ) ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) � R ( f ; n 1 , . . . , n k ) = b (mod n 1 · · · n k ) : � n 1 | f 1 ( b ) , n 2 | f 2 ( b ) , . . . , n k | f k ( b ) f n 1 ··· n k , b ( t ) = f ( n 1 · · · n k t + b ) ∈ Z [ x ] n 1 · · · n k Friable values of polynomials Greg Martin

  81. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Recall notation: prime values of polynomials π ( F ; t ) = # { 1 ≤ n ≤ t : f ( n ) is prime for every irreducible factor f of F } Proposition For f = f 1 . . . f k and x and y sufficiently large, x − b � � � � � M ( f ; x , y ) ∼ · · · π f n 1 ··· n k , b ; . n 1 · · · n k n 1 ≤ x d / y n k ≤ x d / y b ∈R ( f ; n 1 ,..., n k ) ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) � R ( f ; n 1 , . . . , n k ) = b (mod n 1 · · · n k ) : � n 1 | f 1 ( b ) , n 2 | f 2 ( b ) , . . . , n k | f k ( b ) f n 1 ··· n k , b ( t ) = f ( n 1 · · · n k t + b ) ∈ Z [ x ] n 1 · · · n k Friable values of polynomials Greg Martin

  82. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Recall notation: prime values of polynomials π ( F ; t ) = # { 1 ≤ n ≤ t : f ( n ) is prime for every irreducible factor f of F } Proposition For f = f 1 . . . f k and x and y sufficiently large, x − b � � � � � M ( f ; x , y ) ∼ · · · π f n 1 ··· n k , b ; . n 1 · · · n k n 1 ≤ x d / y n k ≤ x d / y b ∈R ( f ; n 1 ,..., n k ) ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) � R ( f ; n 1 , . . . , n k ) = b (mod n 1 · · · n k ) : � n 1 | f 1 ( b ) , n 2 | f 2 ( b ) , . . . , n k | f k ( b ) f n 1 ··· n k , b ( t ) = f ( n 1 · · · n k t + b ) ∈ Z [ x ] n 1 · · · n k Friable values of polynomials Greg Martin

  83. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Recall notation: prime values of polynomials π ( F ; t ) = # { 1 ≤ n ≤ t : f ( n ) is prime for every irreducible factor f of F } Proposition For f = f 1 . . . f k and x and y sufficiently large, x − b � � � � � M ( f ; x , y ) ∼ · · · π f n 1 ··· n k , b ; . n 1 · · · n k n 1 ≤ x d / y n k ≤ x d / y b ∈R ( f ; n 1 ,..., n k ) ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) � R ( f ; n 1 , . . . , n k ) = b (mod n 1 · · · n k ) : � n 1 | f 1 ( b ) , n 2 | f 2 ( b ) , . . . , n k | f k ( b ) f n 1 ··· n k , b ( t ) = f ( n 1 · · · n k t + b ) ∈ Z [ x ] n 1 · · · n k Friable values of polynomials Greg Martin

  84. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Reduction to an arithmetic problem Using Hypothesis H For f = f 1 . . . f k and x and y sufficiently large, x − b � � � � � M ( f ; x , y ) ∼ · · · π f n 1 ··· n k , b ; n 1 · · · n k n 1 ≤ x d / y n k ≤ x d / y b ∈R ( f ; n 1 ,..., n k ) ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) H ( f n 1 ··· n k , b ) x / n 1 · · · n k � � � ∼ · · · ( log x ) k . n 1 ≤ x d / y n k ≤ x d / y b ∈R ( f ; n 1 ,..., n k ) ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) � Next: concentrate on H ( f n 1 ··· n k , b ) b ∈R ( f ; n 1 ,..., n k ) Friable values of polynomials Greg Martin

  85. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Reduction to an arithmetic problem Using Hypothesis H For f = f 1 . . . f k and x and y sufficiently large, x − b � � � � � M ( f ; x , y ) ∼ · · · π f n 1 ··· n k , b ; n 1 · · · n k n 1 ≤ x d / y n k ≤ x d / y b ∈R ( f ; n 1 ,..., n k ) ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) H ( f n 1 ··· n k , b ) x / n 1 · · · n k � � � ∼ · · · ( log x ) k . n 1 ≤ x d / y n k ≤ x d / y b ∈R ( f ; n 1 ,..., n k ) ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) � Next: concentrate on H ( f n 1 ··· n k , b ) b ∈R ( f ; n 1 ,..., n k ) Friable values of polynomials Greg Martin

  86. Introduction Bounds for friable polynomial values Conjecture for friable polynomial values Outline of proof/magic moment Reduction to an arithmetic problem Using Hypothesis H For f = f 1 . . . f k and x and y sufficiently large, x − b � � � � � M ( f ; x , y ) ∼ · · · π f n 1 ··· n k , b ; n 1 · · · n k n 1 ≤ x d / y n k ≤ x d / y b ∈R ( f ; n 1 ,..., n k ) ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) H ( f n 1 ··· n k , b ) x / n 1 · · · n k � � � ∼ · · · ( log x ) k . n 1 ≤ x d / y n k ≤ x d / y b ∈R ( f ; n 1 ,..., n k ) ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) � Next: concentrate on H ( f n 1 ··· n k , b ) b ∈R ( f ; n 1 ,..., n k ) Friable values of polynomials Greg Martin

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