On Zeros of a Polynomial in a Finite Grid: the Alon-Füredi Bound

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1 On Zeros of a Polynomial in a Finite Grid: the Alon-Füredi Bound John R. Schmitt Middlebury College, Vermont, USA joint work with Anurag Bishnoi (Ghent), Pete L. Clark (U. Georgia), Aditya Potukuchi (Rutgers) John R. Schmitt (Middlebury) Alon-Füredi Bound 1 / 21

2 Introduction The strategy [of the polynomial method] is to capture the arbitrary sets of objects (viewed as points in some configuration space) in the zero set of a polynomial whose degree (or other measure of complexity) is under control...one then uses tools from algebraic geometry to understand the structure of this zero set, and thence to control the original sets of objects. Terence Tao, EMS Surveys, 2014 John R. Schmitt (Middlebury) Alon-Füredi Bound 2 / 21

3 Introduction A one variable non-zero polynomial over a field F can have at most as many zeroes as its degree. John R. Schmitt (Middlebury) Alon-Füredi Bound 3 / 21

4 Introduction A one variable non-zero polynomial over a field F can have at most as many zeroes as its degree. Lemma Let F be an arbitrary field, and let f = f (x) be a polynomial in F[x]. Suppose the degree of f is t (thus the x t coefficient of f is nonzero). Then, if A is a subset of F with A > t, there is an a A so that f (a) 0. Example: f (x) = x 2 1 R[x] and A = {1, 1, 7}. f (7) 0. John R. Schmitt (Middlebury) Alon-Füredi Bound 3 / 21

5 Alon s Combinatorial Nullstellensatz Theorem (Non-vanishing corollary to the Combinatorial Nullstellensatz, Noga Alon, 1999) Let F be an arbitrary field, and let f = f (x 1,..., x n ) be a polynomial in F[x 1,..., x n ]. Suppose the degree deg(f ) of f is n i=1 t i, where each t i is a nonnegative integer, and suppose the coefficient of n i=1 x t i i in f is nonzero. Then, if A 1,..., A n are subsets of F with A i > t i, there are a 1 A 1,..., a n A n so that f (a 1,..., a n ) 0. John R. Schmitt (Middlebury) Alon-Füredi Bound 4 / 21

6 Alon s Combinatorial Nullstellensatz Theorem (Chevalley-Warning Theorem) Let n, r, d 1,..., d r Z + with d = d d r < n. For 1 i r, let P i (x 1,..., x n ) F q [x 1,..., x n ] be a polynomial of degree d i. Let Z = Z(P 1,..., P r ) = {a F n q P 1 (a) =... = P r (a) = 0} be the common zero set in F n q of the P i s, and let z = Z. Then: a) (Chevalley s Theorem, 1935) We have z = 0 or z 2. b) (Warning s Theorem, 1935) We have z 0 (mod p). John R. Schmitt (Middlebury) Alon-Füredi Bound 5 / 21

7 Alon s Combinatorial Nullstellensatz Theorem (Chevalley-Warning Theorem) Let n, r, d 1,..., d r Z + with d = d d r < n. For 1 i r, let P i (x 1,..., x n ) F q [x 1,..., x n ] be a polynomial of degree d i. Let Z = Z(P 1,..., P r ) = {a F n q P 1 (a) =... = P r (a) = 0} be the common zero set in F n q of the P i s, and let z = Z. Then: a) (Chevalley s Theorem, 1935) We have z = 0 or z 2. b) (Warning s Theorem, 1935) We have z 0 (mod p). Alon proved (a) using Non-vanishing corollary. Chevalley s Theorem is useful in zero-sum theory. Schauz ( 08) proved (b) using a generalization of Alon s statement. John R. Schmitt (Middlebury) Alon-Füredi Bound 5 / 21

8 The Alon-Füredi Theorem A one variable non-zero polynomial over a field F can have at most as many zeroes as its degree. Lemma Let F be an arbitrary field, and let f = f (x) be a polynomial in F[x]. Suppose the degree of f is t (thus the x t coefficient of f is nonzero). Then, if A is a subset of F with A > t, there is an a A so that f (a) 0. Example: f (x) = x 2 1 R[x] and A = {1, 1, 7, 9, 5}. John R. Schmitt (Middlebury) Alon-Füredi Bound 6 / 21

Let f F[x] = F[x 1,..., x n ] be a polynomial. Let U A = {a A f (a) 0}, u A = U A.

9 The Alon-Füredi Theorem Theorem (Alon-Füredi Theorem, 1993) Let F be a field, let A 1,..., A n be nonempty finite subsets of F. Put A = n i=1 A i for all 1 i n. Let f F[x] = F[x 1,..., x n ] be a polynomial. Let U A = {a A f (a) 0}, u A = U A. Then u A = 0 or u A m( A 1,..., A n ; A A n deg f ). John R. Schmitt (Middlebury) Alon-Füredi Bound 7 / 21

10 The Alon-Füredi Theorem Balls in bins lemma A 1 A 2 A n Bin A i holds at most A i balls. John R. Schmitt (Middlebury) Alon-Füredi Bound 8 / 21

11 The Alon-Füredi Theorem Balls in bins lemma A 1 A 2 A n Bin A i holds at most A i balls. Distribution of N balls is an n-tuple y = (y 1,..., y n ) with y y n = N and 1 y i A i for all i. John R. Schmitt (Middlebury) Alon-Füredi Bound 9 / 21

12 The Alon-Füredi Theorem Balls in bins lemma A 1 A 2 A n Let Π(y) = y 1 y n. If n N A A n, let m( A 1,..., A n ; N) be the minimum value of Π(y) as y ranges over all distributions of N balls into bins A 1,..., A n. John R. Schmitt (Middlebury) Alon-Füredi Bound 10 / 21

13 The Alon-Füredi Theorem Balls in bins lemma A 1 A 2 A n Let Π(y) = y 1 y n. If n N A A n, let m( A 1,..., A n ; N) be the minimum value of Π(y) as y ranges over all distributions of N balls into bins A 1,..., A n. To minimize the product: serve the largest bins first. John R. Schmitt (Middlebury) Alon-Füredi Bound 11 / 21

14 The Alon-Füredi Theorem Theorem (Chevalley-Warning Theorem) Let n, r, d 1,..., d r Z + with d = d d r < n. For 1 i r, let P i (x 1,..., x n ) F q [x 1,..., x n ] be a polynomial of degree d i. Let Z = Z(P 1,..., P r ) = {a F n q P 1 (a) =... = P r (a) = 0} be the common zero set in F n q of the P i s, and let z = Z. Then: a) (Chevalley s Theorem, 1935) We have z = 0 or z 2. b) (Warning s Theorem, 1935) We have z 0 (mod p). Theorem (Warning s Second Theorem) With same hypotheses, z = 0 or z q n d. John R. Schmitt (Middlebury) Alon-Füredi Bound 12 / 21

15 Applications of Alon-Füredi Theorem Encode combinatorial/number-theoretic/incidence-geometry problems via a polynomial so that non-zeros of polynomial correspond to solutions of the given problem. Proof of Warning s Second Theorem: (Clark, Forrow, S ) Let x = (x 1,..., x n ) and P(x) = r (1 P i (x) q 1 ) i=1 P(x) is zero whenever any P i is nonzero. P(x) is nonzero only when each P i is zero. Apply the Alon-Füredi Theorem. John R. Schmitt (Middlebury) Alon-Füredi Bound 13 / 21

16 Applications of Alon-Füredi Theorem Clark, Forrow, S. (14+) showed applications of this to: weighted Davenport constants, generalizations of Erdős-Ginzburg-Ziv Theorem, and graph theory; and Clark (15+) gave strengthenings of these and a further application to: polynomial interpolation. John R. Schmitt (Middlebury) Alon-Füredi Bound 14 / 21

17 Applications of Alon-Füredi Theorem Theorem (Schwartz-Zippel Lemma, 1979) Let R be a domain and let S R be finite and nonempty, with S := s. Let f R[x 1,..., x n ] be a nonzero polynomial. Then z S n(f ) (deg f )s n 1. (1) John R. Schmitt (Middlebury) Alon-Füredi Bound 15 / 21

18 Applications of Alon-Füredi Theorem Theorem (Schwartz-Zippel Lemma, 1979) Let R be a domain and let S R be finite and nonempty, with S := s. Let f R[x 1,..., x n ] be a nonzero polynomial. Then Proof. The conclusion is equivalent to z S n(f ) (deg f )s n 1. (1) u S n(f ) s n 1 (s deg f ). John R. Schmitt (Middlebury) Alon-Füredi Bound 15 / 21

19 Applications of Alon-Füredi Theorem Theorem (Schwartz-Zippel Lemma, 1979) Let R be a domain and let S R be finite and nonempty, with S := s. Let f R[x 1,..., x n ] be a nonzero polynomial. Then Proof. The conclusion is equivalent to z S n(f ) (deg f )s n 1. (1) u S n(f ) s n 1 (s deg f ). If deg f s, then (1) asserts that f has no more zeros on S n than the size of S n : true. So the nontrivial case is deg f < s. John R. Schmitt (Middlebury) Alon-Füredi Bound 15 / 21

20 Applications of Alon-Füredi Theorem Theorem (Schwartz-Zippel Lemma, 1979) Let R be a domain and let S R be finite and nonempty, with S := s. Let f R[x 1,..., x n ] be a nonzero polynomial. Then Proof. The conclusion is equivalent to z S n(f ) (deg f )s n 1. (1) u S n(f ) s n 1 (s deg f ). If deg f s, then (1) asserts that f has no more zeros on S n than the size of S n : true. So the nontrivial case is deg f < s. Then apply Alon-Füredi, so u S n(f ) m(s,..., s; ns deg f ) = s n 1 (s deg f ). John R. Schmitt (Middlebury) Alon-Füredi Bound 15 / 21

21 Applications of Alon-Füredi Theorem Theorem (DeMillo-Lipton-Zippel Theorem, 1978) Let R be a domain, let f R[x 1,..., x n ] be a nonzero polynomial, and let d Z + be such that deg xi f d for all i [n]. Let S R be a nonempty set with S := s > d elements. Then z S n(f ) s n (s d) n. John R. Schmitt (Middlebury) Alon-Füredi Bound 16 / 21

22 Applications of Alon-Füredi Theorem Example Let S be a finite subset of R containing 0 and of size s 3. Let f = x 1 x 2 R[x 1, x 2 ]. Then we have DeMillo-Lipton-Zippel gives Schwartz-Zippel gives The Alon-Füredi Theorem gives z S 2(f ) = 2s 1. z S 2(f ) s 2 (s 1) 2 = 2s 1. z S 2(f ) 2s. z S 2(f ) s 2 m(s, s; 2s 2) = s 2 s(s 2) = 2s. Thus neither Alon-Füredi nor Schwartz-Zippel implies DeMillo-Lipton-Zippel. John R. Schmitt (Middlebury) Alon-Füredi Bound 17 / 21

23 Applications of Alon-Füredi Theorem Example For the other direction, take f = x 1 + x 2. DeMillo-Lipton-Zippel gives z S 2(f ) s 2 (s 1) 2 = 2s 1. Schwartz-Zippel and Alon-Füredi give z S 2(f ) s. Thus DeMillo-Lipton-Zippel does not imply Schwartz-Zippel or Alon-Füredi. John R. Schmitt (Middlebury) Alon-Füredi Bound 18 / 21

24 Applications of Alon-Füredi Theorem Theorem (Generalized Alon-Füredi Theorem; A. Bishnoi, P.L. Clark, A. Potukuchi, S (15+)) Let R be a ring and let A 1,..., A n be non-empty finite subsets of R that satisfy Condition (D). For i [n], let b i be an integer such that 1 b i A i. Let f R[x 1,..., x n ] be a non-zero polynomial such that deg xi f A i b i for all i [n]. Let U A = {a A : f (a) 0} where A = A 1 A n R n. Then we have u A m( A 1,..., A n ; b 1,..., b n ; Moreover, this bound is sharp in all cases. n A i deg f ). i=1 John R. Schmitt (Middlebury) Alon-Füredi Bound 19 / 21

25 Applications of Alon-Füredi Theorem Theorem (Generalized Alon-Füredi Theorem; A. Bishnoi, P.L. Clark, A. Potukuchi, S (15+)) Let R be a ring and let A 1,..., A n be non-empty finite subsets of R that satisfy Condition (D). For i [n], let b i be an integer such that 1 b i A i. Let f R[x 1,..., x n ] be a non-zero polynomial such that deg xi f A i b i for all i [n]. Let U A = {a A : f (a) 0} where A = A 1 A n R n. Then we have u A m( A 1,..., A n ; b 1,..., b n ; Moreover, this bound is sharp in all cases. n A i deg f ). i=1 Generalized Alon-Füredi does imply DeMillo-Lipton-Zippel. John R. Schmitt (Middlebury) Alon-Füredi Bound 19 / 21

26 Applications of Alon-Füredi Theorem Theorem (Generalized Alon-Füredi Theorem; A. Bishnoi, P.L. Clark, A. Potukuchi, S (15+)) Let R be a ring and let A 1,..., A n be non-empty finite subsets of R that satisfy Condition (D). For i [n], let b i be an integer such that 1 b i A i. Let f R[x 1,..., x n ] be a non-zero polynomial such that deg xi f A i b i for all i [n]. Let U A = {a A : f (a) 0} where A = A 1 A n R n. Then we have u A m( A 1,..., A n ; b 1,..., b n ; Moreover, this bound is sharp in all cases. n A i deg f ). i=1 Generalized Alon-Füredi does imply DeMillo-Lipton-Zippel. (Generalized) Alon-Füredi has other applications to coding theory and finite geometry. John R. Schmitt (Middlebury) Alon-Füredi Bound 19 / 21

27 Applications of Alon-Füredi Theorem A nonempty subset S R is said to satisfy Condition (D) if for all x y S, the element x y R is not a zero divisor. A finite grid is a subset A = n i=1 A i of R n (for some n Z + ) with each A i a finite, nonempty subset of R. We say that A satisfies Condition (D) if each A i does. Given any b 1,..., b n Z with 1 b i a i, we may consider the scenario in which the i-th bin comes prefilled with b i balls. If n i=1 b i N n i=1 a i, we may restrict to distributions y = (y 1,..., y n ) of N balls into bins of sizes a 1,..., a n such that b i y i a i for all i [n] and put m(a 1,..., a n ; b 1,..., b n ; N) = min Π(y), where the minimum ranges over this restricted set of distributions. John R. Schmitt (Middlebury) Alon-Füredi Bound 20 / 21

28 Applications of Alon-Füredi Theorem Alon, Noga Combinatorial Nullstellensatz. Recent trends in combinatorics (Mátraháza, 1995). Combin. Probab. Comput. 8 (1999), no. 1-2, P. L. Clark, A. Forrow and J.R.S., Warning s Second Theorem With Restricted Variables. To appear in Combinatorica, (2014) R. Lipton, The curious history of the Schwartz-Zippel Lemma. Tao, Terence Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory. EMS Surv. Math. Sci. 1 (2014), no. 1, Vielen Dank! John R. Schmitt (Middlebury) Alon-Füredi Bound 21 / 21

ON ZEROS OF A POLYNOMIAL IN A FINITE GRID ANURAG BISHNOI. Department Mathematics, Ghent University, Belgium PETE L. CLARK

ON ZEROS OF A POLYNOMIAL IN A FINITE GRID ANURAG BISHNOI Department Mathematics, Ghent University, Belgium PETE L. CLARK Department of Mathematics, University of Georgia, Athens, GA, USA ADITYA POTUKUCHI