Quantitative Equidistribution for the Solutions of Systems of Sparse - PowerPoint PPT Presentation

Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations Carlos D’Andrea SIAM Conference on Applied Algebraic Geometry – Raleigh NC October 2011 Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

Roots of polynomials vs “controlled” coefficients Let f be a polynomial of degree d ≫ 0 with coefficients in {− 1 , 0 , 1 } . I will plot all complex solutions of f = 0 . . . Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

For instance, let d = 10 and f = − x 10 + x 9 + x 8 + x 6 + x 5 − x 4 + x 3 − x 2 + x − 1 Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

Now set d = 30 and = x 30 − x 29 − x 28 + x 26 + x 25 − x 24 − x 23 − x 22 f + x 21 − x 20 + x 19 + x 18 + x 16 + x 15 − x 14 + x 13 + x 12 + x 10 + x 9 − x 6 + x 5 − 1 Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

d = 100 and f = − x 100 − x 98 + x 96 + x 94 − x 93 + x 92 − x 91 − x 90 + · · Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

A more ambitious experiment Let us say now that f has degree d ≫ 0 with integer coefficients between − d and d . What happens now? Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

d = 10 and f = − 6 + 8 x − x 2 + 10 x 3 − 3 x 4 + 8 x 5 + 4 x 6 − 9 x 7 + 9 x 8 − 6 x 9 + 5 x 10 Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

d = 50 and f = − 24 + 12 x − 44 x 48 − 48 x 49 − 42 x 28 + 15 x 29 + 34 x 26 + 22 x 27 − 24 x 24 + 29 x 25 + 14 x 2 − 40 x 3 − 48 x 4 + 35 x 5 + 24 x 6 + 27 x 7 − 3 x 8 − 15 x 9 − 21 x 10 + 12 x 14 − 15 x 50 − 14 x 33 + 38 x 34 + 10 x 35 − 23 x 36 + 48 x 37 + 30 x 38 − 23 x 39 − 31 x 40 + 2 x 41 + 24 x 42 + 9 x 43 − 15 x 44 − 29 x 45 + 45 x 46 + 40 x 47 + 40 x 31 − 40 x 32 + 38 x 11 + 8 x 12 − 16 x 13 − 39 x 15 + 2 x 16 − 38 x 17 − x 18 + 16 x 19 − 44 x 20 − 20 x 21 + 22 x 22 + 28 x 23 + 32 x 30 Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

d = 100 and f = 30 − 45 x − 91 x 74 − 33 x 75 + 4 x 73 − 59 x 79 + 35 x 92 − 57 x 48 + 49 x 49 + 2 x 93 − 87 x 28 − 16 x 29 − 78 x 26 − 31 x 27 + 19 x 50 − 73 x 24 − 63 x 25 + 98 x 2 + 29 x 3 − 97 x 4 + 47 x 5 + 46 x 6 − 88 x 7 − 74 x 8 − 60 x 9 − 62 x 10 − 27 x 81 − 82 x 80 − 92 x 78 − 50 x 77 − 41 x 76 − 21 x 95 + 8 x 66 − 7 x 67 + 75 x 64 − 19 x 94 − 48 x 63 + 92 x 65 − 18 x 60 + 53 x 61 + 84 x 59 − 15 x 57 − 13 x 58 − 64 x 91 + 84 x 90 − 54 x 89 + 67 x 55 − 81 x 56 − 27 x 54 − 61 x 88 + 43 x 87 + 49 x 86 + 51 x 84 − 12 x 85 − 64 x 83 + 52 x 82 + 43 x 70 − 91 x 71 − 97 x 72 + 76 x 68 + 14 x 69 + 73 x 99 − 56 x 97 + 41 x 98 + 73 x 96 + 44 x 100 + 2 x 51 − 79 x 52 + 87 x 53 − 43 x 14 + 39 x 62 + 50 x 33 + 53 x 34 + 64 x 35 + 57 x 36 − 57 x 37 − 31 x 38 + 85 x 39 + 30 x 40 − 49 x 41 + 6 x 42 − 82 x 43 + 34 x 44 + 59 x 45 + 7 x 46 + 91 x 47 + 59 x 31 + 58 x 32 − 4 x 11 − 71 x 12 − 68 x 13 + 74 x 15 + 60 x 16 − 3 x 17 + 23 x 18 − 55 x 19 + 80 x 20 − 32 x 21 + 17 x 22 − 14 x 23 − 69 x 30 Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

What is going on??? Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

The Erdös-Turán theorem Let f ( x ) = a d x d + · · · + a 0 = a d ( x − ρ 1 e i θ 1 ) · · · ( x − ρ d e i θ d ) Definition The angle discrepancy of f is � # { k : α ≤ θ k < β } − β − α � � � ∆ θ ( f ) := sup � � d 2 π � � 0 ≤ α<β< 2 π The ε -radius discrepancy of f is ∆ r ( f ; ε ) := 1 1 � � k : 1 − ε < ρ k < d # 1 − ε Also set || f || := sup | z | = 1 | f ( z ) | Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

Theorem [Erdös-Turán 1948], [Hughes-Nikeghbali 2008] � � � � � || f || || f || 1 2 √ √ ∆ θ ( f ) ≤ c d log 1 − ∆ r ( f ; ε ) ≤ ε d log , | a 0 a d | | a 0 a d | √ Here 2 ≤ c ≤ 2 , 5619 [Amoroso-Mignotte 1996] Corollary: the equidistribution || f d || � √ � Let f d ( x ) of degree d such that log = o ( d ) , then | a d , 0 a d , d | 1 = β − α � � lim d →∞ d # k : α ≤ θ dk < β 2 π 1 1 � � lim d →∞ d # k : 1 − ε < ρ dk < = 1 1 − ε Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

Some consequences � � � || f || √ 1 The number of real roots of f is ≤ 51 d log | a 0 a d | [Erhardt-Schur-Szego] 2 If g ( z ) = 1 + b 1 z + b 2 z 2 + . . . converges on the unit disk, then the zeros of its d -partial sums distribute uniformely on the unit circle as d → ∞ [Jentzsch-Szego] Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

Equidistribution in several variables (joint work with Martin Sombra & André Galligo) * For a finite sequence of points P = { p 1 , . . . , p m } ⊂ ( C × ) n , we can define ∆ θ ( P ) and ∆ r ( P , ε ) * Every such set P is the solution set of a complete intersection f = 0 with f = ( f 1 , . . . , f n ) Laurent Polynomials in C [ x ± 1 1 , . . . , x ± 1 n ] Problem * Estimate ∆ θ ( P ) and ∆ r ( P , ε ) in terms of f || f || √ * Which is the analogue of | a 0 a d | in several variables? * Equidistribution theorems Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

Some Evidence Singularities of families of algebraic plane curves with “controlled” coefficients tend to the equidistribution [Diaconis-Galligo] Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

More Evidence: equidistribution of algebraic points A sequence of algebraic points { p k } k ∈ N ⊂ ( C ∗ ) n such that deg ( p k ) = k and lim k →∞ h ( p k ) = 0 “equidistributes” in S 1 × S 1 × . . . × S 1 [Bilu 1997] Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

The multivariate setting For f 1 , . . . , f n ∈ C [ x ± 1 1 , . . . , x ± 1 n ] consider V ( f 1 , . . . , f n ) = { ξ ∈ ( C × ) n : f 1 ( ξ ) = · · · = f n ( ξ ) = 0 } ⊂ ( C × ) n and V 0 the subset of isolated points Set Q i := N ( f i ) ⊂ R n the Newton polytope, then # V 0 ≤ MV n ( Q 1 , . . . , Q n ) =: D [ BKK ] From now on, we will assume # V 0 = D , in particular V ( f ) = V 0 . Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

A toric variety in the background # V 0 = D is equivalent to the fact that the system f 1 = 0 , . . . , f n = 0 does not have solutions in the toric variety associated to the polytope Q 1 + Q 2 + . . . + Q n [Bernstein 1975], [Huber-Sturmfels 1995] Can be tested with resultants “at infinity”! Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

A multivariate Erdös-Turán measure f ↔ “multidirectional” Chow forms a 0 , a d ↔ facet resultants E f , a ( z ) = Res { 0 , a } , A 1 ,..., A n ( z − x a , f 1 , . . . , f n ) � � � E f , a ( z ) � 1 η ( f ) = sup a ∈ Z n \{ 0 } D � a � log |� v , a �| n ( f v 1 ,..., f v � v | Res A v n ) | 2 1 ,..., A v Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

Theorem (D-Galligo-Sombra) η ( f ) < + ∞ For n = 1 , η ( f ) coincides with the Erdös-Turán measure � f � √ | a 0 a D | If f 1 , . . . , f n ∈ C [ x ± 1 1 , . . . , x ± 1 n ] and f = 0 has D > 0 zeroes, then � 1 � 1 3 log + ∆ θ ( f ) ≤ c ( n ) η ( f ) , 1 − ∆ r ( f ; ε ) ≤ c ( n ) η ( f ) η ( f ) n + 1 with c ( n ) ≤ 2 3 n n 2 Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

Corollary (D-Galligo-Sombra) The number of real roots of a sparse system f = 0 with f 1 , . . . , f n ∈ R [ x ± 1 1 , . . . , x ± 1 n ] is bounded above by � 1 � 1 D c ′ ( n ) η ( f ) 3 log + η ( f ) n + 1 with c ′ ( n ) ≤ 2 4 n n 2 Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

Estimates on η ( f ) Suppose Q i ⊂ d i ∆ + a i , with ∆ being the fundamental simplex of R n . Then � log � f j � sup � n 1 η ( f ) 2 nd 1 . . . d n + < j = 1 D d j v � v � log + | Res A v 1 n ( f v 1 , . . . , f v n ) − 1 | � � 1 ,..., A v 2 In particular, for f 1 , . . . , f n ∈ Z [ x 1 , . . . , x n ] , of degrees d 1 , . . . , d n , then n log � f j � sup � η ( f ) ≤ 2 n d j j = 1 Carlos D’Andrea Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations