Two Ways to Count Solutions to Polynomial Robinson Equations - PowerPoint PPT Presentation

Two Ways to Count Solutions to Polynomial Equations Margaret Two Ways to Count Solutions to Polynomial Robinson Equations Margaret Robinson Mount Holyoke College May 24, 2013

Two Ways to Count Generating functions Solutions to Polynomial Equations A generating function is a clothesline Margaret Robinson on which we hang up a sequence of numbers for display.—Herbert Wilf Given a sequence of numbers a 0 , a 1 , a 2 , .... we can form its generating function ∞ � a n t n f ( t ) = n =0

Rational Generating Functions Two Ways to Count Solutions to Using formulas like Polynomial Equations ∞ 1 Margaret t n = � Robinson 1 − t , n =0 ∞ 1 ( n + 1) t n = � (1 − t ) 2 n =0 and ∞ ( n + 1)( n + 2) 1 t n = � (1 − t ) 3 , 2 n =0 Some generating functions can be seen to be rational functions of t !

First Generating Function Two Ways to Count Solutions to Consider a prime number p and a polynomial Polynomial Equations f ( x ) = f ( x 1 , ..., x n ) in n variables with Margaret Robinson coefficients in Z and consider f with coefficients reduced modulo p .

First Generating Function Two Ways to Count Solutions to Consider a prime number p and a polynomial Polynomial Equations f ( x ) = f ( x 1 , ..., x n ) in n variables with Margaret Robinson coefficients in Z and consider f with coefficients reduced modulo p . Let | N e | = Card { x ∈ F ( n ) p e | f ( x ) = 0 in F p e } .

First Generating Function Two Ways to Count Solutions to Consider a prime number p and a polynomial Polynomial Equations f ( x ) = f ( x 1 , ..., x n ) in n variables with Margaret Robinson coefficients in Z and consider f with coefficients reduced modulo p . Let | N e | = Card { x ∈ F ( n ) p e | f ( x ) = 0 in F p e } . Define the Weil Poincar´ e Series as: ∞ � | N e | t e P Weil ( t ) = e =0 with | N 0 | = 1 and | N e | ≤ p ne .

Second Generating Function Two Ways to Count Solutions to Consider a prime number p and a polynomial Polynomial Equations f ( x ) = f ( x 1 , ..., x n ) in n variables with Margaret Robinson coefficients in Z and for x ∈ Z ( n ) .

Second Generating Function Two Ways to Count Solutions to Consider a prime number p and a polynomial Polynomial Equations f ( x ) = f ( x 1 , ..., x n ) in n variables with Margaret Robinson coefficients in Z and for x ∈ Z ( n ) . Let | N d | = Card { x mod p d | f ( x ) ≡ 0 mod p d } .

Second Generating Function Two Ways to Count Solutions to Consider a prime number p and a polynomial Polynomial Equations f ( x ) = f ( x 1 , ..., x n ) in n variables with Margaret Robinson coefficients in Z and for x ∈ Z ( n ) . Let | N d | = Card { x mod p d | f ( x ) ≡ 0 mod p d } . Define the Igusa Poincar´ e Series as: ∞ � | N d | t d P Igusa ( t ) = d =0 with | N 0 | = 1 and | N d | ≤ p nd .

Two Ways to Both these generating functions are known to be Count Solutions to Polynomial rational functions of t . Equations Margaret Robinson

Two Ways to Both these generating functions are known to be Count Solutions to Polynomial rational functions of t . Equations Margaret Robinson Theorem (Dwork, 1959) P Weil ( t ) is a rational function of t . | N e | = � u i − � v i =1 α e i =1 β e i (Special case of the first part of the Weil Conjectures 1949.)

Two Ways to Both these generating functions are known to be Count Solutions to Polynomial rational functions of t . Equations Margaret Robinson Theorem (Dwork, 1959) P Weil ( t ) is a rational function of t . | N e | = � u i − � v i =1 α e i =1 β e i (Special case of the first part of the Weil Conjectures 1949.) Theorem (Igusa, 1975) P Igusa ( t ) is a rational function of t . (Conjectured in exercises of the 1966 textbook by Borevich and Shafarevich.)

Two Ways to Example 1 Count Solutions to Polynomial Let Equations Margaret f ( x ) = x Robinson Then | N e | = | N d | = 1 . Hence,

Two Ways to Example 1 Count Solutions to Polynomial Let Equations Margaret f ( x ) = x Robinson Then | N e | = | N d | = 1 . Hence, ∞ t e = � P Weil ( t ) = P Igusa ( t ) = e =0

Two Ways to Example 1 Count Solutions to Polynomial Let Equations Margaret f ( x ) = x Robinson Then | N e | = | N d | = 1 . Hence, ∞ 1 t e = � P Weil ( t ) = P Igusa ( t ) = (1 − t ) e =0

Two Ways to Example 2 Count Solutions to Polynomial Let Equations Margaret f ( x , y ) = xy Robinson Then | N e | = 2 p e − 1 . Hence,

Two Ways to Example 2 Count Solutions to Polynomial Let Equations Margaret f ( x , y ) = xy Robinson Then | N e | = 2 p e − 1 . Hence, ∞ (2 p e − 1) t e = � P Weil ( t ) = e =0

Two Ways to Example 2 Count Solutions to Polynomial Let Equations Margaret f ( x , y ) = xy Robinson Then | N e | = 2 p e − 1 . Hence, ∞ 1 + ( p − 2) t (2 p e − 1) t e = � P Weil ( t ) = (1 − t )(1 − pt ) e =0

Two Ways to Example 2 (continued) Count Counting points solutions of f ( x , y ) = xy mod p d for each Solutions to Polynomial Equations d , we see that | N d | is more complicated but we find the Margaret recursion relation: Robinson | N 0 | = 1

Two Ways to Example 2 (continued) Count Counting points solutions of f ( x , y ) = xy mod p d for each Solutions to Polynomial Equations d , we see that | N d | is more complicated but we find the Margaret recursion relation: Robinson | N 0 | = 1 | N 1 | = 2 p − 1

Two Ways to Example 2 (continued) Count Counting points solutions of f ( x , y ) = xy mod p d for each Solutions to Polynomial Equations d , we see that | N d | is more complicated but we find the Margaret recursion relation: Robinson | N 0 | = 1 | N 1 | = 2 p − 1 p ( | N 1 | − 1) + p 2 | N 0 | = 3 p 2 − 2 p | N 2 | =

Two Ways to Example 2 (continued) Count Counting points solutions of f ( x , y ) = xy mod p d for each Solutions to Polynomial Equations d , we see that | N d | is more complicated but we find the Margaret recursion relation: Robinson | N 0 | = 1 | N 1 | = 2 p − 1 p ( | N 1 | − 1) + p 2 | N 0 | = 3 p 2 − 2 p | N 2 | = p d − 1 ( | N 1 | − 1) + p 2 | N d − 2 | | N d | = With careful counting and induction we get the closed form expression:

Two Ways to Example 2 (continued) Count Counting points solutions of f ( x , y ) = xy mod p d for each Solutions to Polynomial Equations d , we see that | N d | is more complicated but we find the Margaret recursion relation: Robinson | N 0 | = 1 | N 1 | = 2 p − 1 p ( | N 1 | − 1) + p 2 | N 0 | = 3 p 2 − 2 p | N 2 | = p d − 1 ( | N 1 | − 1) + p 2 | N d − 2 | | N d | = With careful counting and induction we get the closed form expression: | N d | = ( d + 1) p d − dp d − 1

Example 2 (continued) Two Ways to Count The Igusa Poincar´ e series for the polynomial f ( x , y ) = xy Solutions to Polynomial is: Equations ∞ Margaret [( d + 1) p d − dp d − 1 ] t d � Robinson P Igusa ( t ) = d =0

Example 2 (continued) Two Ways to Count The Igusa Poincar´ e series for the polynomial f ( x , y ) = xy Solutions to Polynomial is: Equations ∞ Margaret [( d + 1) p d − dp d − 1 ] t d � Robinson P Igusa ( t ) = d =0 ∞ ( d + 1)( pt ) d − dp − 1 ( pt ) d � = 1 + d =1

Example 2 (continued) Two Ways to Count The Igusa Poincar´ e series for the polynomial f ( x , y ) = xy Solutions to Polynomial is: Equations ∞ Margaret [( d + 1) p d − dp d − 1 ] t d � Robinson P Igusa ( t ) = d =0 ∞ ( d + 1)( pt ) d − dp − 1 ( pt ) d � = 1 + d =1 ∞ ∞ d (1 − p − 1 )( pt ) d + � � ( pt ) d = 1 + d =1 d =1

Example 2 (continued) Two Ways to Count The Igusa Poincar´ e series for the polynomial f ( x , y ) = xy Solutions to Polynomial is: Equations ∞ Margaret [( d + 1) p d − dp d − 1 ] t d � Robinson P Igusa ( t ) = d =0 ∞ ( d + 1)( pt ) d − dp − 1 ( pt ) d � = 1 + d =1 ∞ ∞ d (1 − p − 1 )( pt ) d + � � ( pt ) d = 1 + d =1 d =1 1 + (1 − p − 1 )( pt ) pt = + (1 − pt ) 2 (1 − pt )

Example 2 (continued) Two Ways to Count The Igusa Poincar´ e series for the polynomial f ( x , y ) = xy Solutions to Polynomial is: Equations ∞ Margaret [( d + 1) p d − dp d − 1 ] t d � Robinson P Igusa ( t ) = d =0 ∞ ( d + 1)( pt ) d − dp − 1 ( pt ) d � = 1 + d =1 ∞ ∞ d (1 − p − 1 )( pt ) d + � � ( pt ) d = 1 + d =1 d =1 1 + (1 − p − 1 )( pt ) pt = + (1 − pt ) 2 (1 − pt ) 1 − t = (1 − pt ) 2

Two Ways to Count Solutions to Example 3 Polynomial Equations Let Margaret f ( x , y ) = y 2 − x 3 Robinson ∞ P Igusa ( p − 2 t ) = � | N d | ( p − 2 t ) d d =0

Two Ways to Count Solutions to Example 3 Polynomial Equations Let Margaret f ( x , y ) = y 2 − x 3 Robinson ∞ P Igusa ( p − 2 t ) = � | N d | ( p − 2 t ) d d =0 = (1 + p − 2 t 2 − p − 3 t 2 − p − 6 t 6 ) (1 − p − 1 t )(1 − p − 5 t 6 )

Example 3 (continued) Two Ways to Count Solutions to From the Igusa Poincar´ e series for Polynomial Equations f ( x , y ) = y 2 − x 3 , we get a recursion relation of Margaret Robinson the form: | N 0 | = 1

Example 3 (continued) Two Ways to Count Solutions to From the Igusa Poincar´ e series for Polynomial Equations f ( x , y ) = y 2 − x 3 , we get a recursion relation of Margaret Robinson the form: | N 0 | = 1 | N 1 | = p