The Multivariate Schwartz-Zippel Lemma M. Levent Do gan Joint work - PowerPoint PPT Presentation

Introduction and the Main Theorem Applications The Algorithm The Multivariate Schwartz-Zippel Lemma M. Levent Do˘ gan Joint work with A. A. Erg¨ ur, J. D. Mundo and E. Tsigaridas Technische Universit¨ at Berlin EuroCG 2020 W¨ urzburg - 18.03.2020

Introduction and the Main Theorem Applications The Algorithm Table of Contents Introduction and the Main Theorem Applications The Algorithm

Introduction and the Main Theorem Applications The Algorithm There is a wide literature on counting number of zeroes of a polynomial on a finite grid thanks to its applications to Polynomial Identity Testing, Incidence Geometry and Extremal Combinatorics. Theorem (The Schwartz-Zippel-DeMillo-Lipton Lemma) Let F be a field, let S ⊆ F be a finite set and let 0 � = p ∈ F [ x 1 , x 2 , . . . , x n ] be a polynomial of degree d . Suppose | S | > d and let S n := S × S × · · · × S . Then we have | Z ( p ) ∩ S n | ≤ d | S | n − 1 where Z ( p ) = { v ∈ F n | p ( v ) = 0 } denotes the zero locus of p.

Introduction and the Main Theorem Applications The Algorithm There is a wide literature on counting number of zeroes of a polynomial on a finite grid thanks to its applications to Polynomial Identity Testing, Incidence Geometry and Extremal Combinatorics. Theorem (The Schwartz-Zippel-DeMillo-Lipton Lemma) Let F be a field, let S ⊆ F be a finite set and let 0 � = p ∈ F [ x 1 , x 2 , . . . , x n ] be a polynomial of degree d . Suppose | S | > d and let S n := S × S × · · · × S . Then we have | Z ( p ) ∩ S n | ≤ d | S | n − 1 where Z ( p ) = { v ∈ F n | p ( v ) = 0 } denotes the zero locus of p. A theorem on the same direction is given by Alon: Theorem (Alon’s Combinatorial Nullstellensatz) Let p ∈ F [ x 1 , x 2 , . . . , x n ] be a polynomial of degree d = � n i =1 d i for some positive i =1 x d i integers d i and assume that the coefficient of the monomial � n in p is non-zero. i Let S i ⊆ F be finite sets with | S i | > d i and let S := S 1 × S 2 × · · · × S n . Then, there exists v ∈ S such that p ( v ) � = 0 .

Introduction and the Main Theorem Applications The Algorithm In this talk, we want to obtain similar results for multi-grids . Notation We call a sequence λ = ( λ 1 , λ 2 , . . . , λ m ) of positive integers a partition of n into m parts if n = λ 1 + λ 2 + · · · + λ m . In this case, we write λ ⊢ m n . Given a partition λ ⊢ m n , we introduce the notation x 1 = ( x 1 , x 2 , . . . , x λ 1 ) , x 2 = ( x λ 1 +1 , x λ 1 +2 , . . . , x λ 1 + λ 2 ) and so on. Given finite sets S 1 ⊆ F λ 1 , S 2 ⊆ F λ 2 , . . . , S m ⊆ F λ m , we call the product S := S 1 × S 2 × · · · × S m the multi-grid defined by S 1 , S 2 , . . . , S m .

Introduction and the Main Theorem Applications The Algorithm In this talk, we want to obtain similar results for multi-grids . Notation We call a sequence λ = ( λ 1 , λ 2 , . . . , λ m ) of positive integers a partition of n into m parts if n = λ 1 + λ 2 + · · · + λ m . In this case, we write λ ⊢ m n . Given a partition λ ⊢ m n , we introduce the notation x 1 = ( x 1 , x 2 , . . . , x λ 1 ) , x 2 = ( x λ 1 +1 , x λ 1 +2 , . . . , x λ 1 + λ 2 ) and so on. Given finite sets S 1 ⊆ F λ 1 , S 2 ⊆ F λ 2 , . . . , S m ⊆ F λ m , we call the product S := S 1 × S 2 × · · · × S m the multi-grid defined by S 1 , S 2 , . . . , S m . Given a multivariate polynomial p ∈ C [ x 1 , x 2 , . . . , x m ], we want to bound number of zeros of p can have on a multi-grid S . It turns out that this task is impossible without imposing some conditions for p .

Introduction and the Main Theorem Applications The Algorithm Example Let g 1 ∈ C [ x 1 , x 2 ] \ C and g 2 ∈ C [ x 3 , x 4 ] \ C . For h 1 , h 2 ∈ C [ x 1 , x 2 , x 3 , x 4 ], set p = g 1 h 1 + g 2 h 2 . Observe that Z ( g 1 ) and Z ( g 2 ) are planar curves in C 2 and Z ( p ) contains Z ( g 1 ) × Z ( g 2 ). In particular, p can vanish on arbitrarily large Cartesian products!

Introduction and the Main Theorem Applications The Algorithm Example Let g 1 ∈ C [ x 1 , x 2 ] \ C and g 2 ∈ C [ x 3 , x 4 ] \ C . For h 1 , h 2 ∈ C [ x 1 , x 2 , x 3 , x 4 ], set p = g 1 h 1 + g 2 h 2 . Observe that Z ( g 1 ) and Z ( g 2 ) are planar curves in C 2 and Z ( p ) contains Z ( g 1 ) × Z ( g 2 ). In particular, p can vanish on arbitrarily large Cartesian products! Definition m n . An affine variety V ⊆ C n is called λ -reducible if there exist Let λ ⊢ positive dimensional varieties V i ⊆ C λ i such that V 1 × V 2 × · · · × V m ⊆ V . Otherwise, we say V is λ -irreducible. A polynomial p ∈ C [ x 1 , x 2 , . . . , x n ] is said to be λ -reducible (resp. λ -irreducible) if the hypersurface Z ( p ) defined by p is λ -reducible (resp. λ -irreducible).

Introduction and the Main Theorem Applications The Algorithm The Main Theorem Theorem (D., Erg¨ ur, Mundo, Tsigaridas) Let λ ⊢ m n be a partition of n into m parts and let p ∈ C [ x 1 , x 2 , . . . , x n ] be a λ -irreducible polynomial of degree d ≥ 2 . Let S i ⊆ C λ i and let S := S 1 × S 2 × · · · × S m be the multi-grid defined by S i . Then, for all ε > 0 , we have m m λ i +1 + ε + d 2 n 4 1 1 − | Z ( p ) ∩ S | = O n ,ε ( d 5 � � � | S i | | S j | ) i =1 i =1 j � = i where O n ,ε notation only hides constants depending on n and ε .

Introduction and the Main Theorem Applications The Algorithm The Main Theorem Theorem (D., Erg¨ ur, Mundo, Tsigaridas) Let λ ⊢ m n be a partition of n into m parts and let p ∈ C [ x 1 , x 2 , . . . , x n ] be a λ -irreducible polynomial of degree d ≥ 2 . Let S i ⊆ C λ i and let S := S 1 × S 2 × · · · × S m be the multi-grid defined by S i . Then, for all ε > 0 , we have m m λ i +1 + ε + d 2 n 4 1 1 − | Z ( p ) ∩ S | = O n ,ε ( d 5 � � � | S i | | S j | ) i =1 i =1 j � = i where O n ,ε notation only hides constants depending on n and ε . Observation As long as we check λ -irreducibility over C , the bound works over any subfield of C .

Introduction and the Main Theorem Applications The Algorithm Table of Contents Introduction and the Main Theorem Applications The Algorithm

Introduction and the Main Theorem Applications The Algorithm Point-Line Incidences Theorem (Szemer´ edi-Trotter) Let P be a set of points and L be a set of lines in the real plane, R 2 . Let I ( P , L ) = { ( p , l ) ∈ P × L | p ∈ l } be the set of incidences between P and L. Then |I ( P , L ) | = O ( | P | 2 / 3 | L | 2 / 3 + | P | + | L | ) .

Introduction and the Main Theorem Applications The Algorithm Point-Line Incidences Theorem (Szemer´ edi-Trotter) Let P be a set of points and L be a set of lines in the real plane, R 2 . Let I ( P , L ) = { ( p , l ) ∈ P × L | p ∈ l } be the set of incidences between P and L. Then |I ( P , L ) | = O ( | P | 2 / 3 | L | 2 / 3 + | P | + | L | ) . The theorem holds if we replace R 2 with C 2 . To our knowledge, the complex version is first proven by T´ oth. As our first application, we use the main theorem to recover the above bound, except for ε in the exponent: Theorem (Cheap Szemer´ edi-Trotter Theorem) Let P be a set of points and L be a set of lines in C 2 (or R 2 ). Then, for any ε > 0 , there are at most O ( | P | 2 / 3+ ε | L | 2 / 3+ ε + | P | + | L | ) incidences between P and L.

Introduction and the Main Theorem Applications The Algorithm Proof. Let p = x 1 + x 2 x 3 + x 4 ∈ C [ x 1 , x 2 , x 3 , x 4 ]. It is straightforward to show that p is (2 , 2)-irreducible: For u = ( u 1 , u 2 ) , v = ( v 1 , v 2 ) ∈ C 2 , the equations p ( u 1 , u 2 , x 3 , x 4 ) = 0 , p ( v 1 , v 2 , x 3 , x 4 ) = 0 are (affine) linear in x 3 , x 4 , thus has at most one solution. We deduce that Z ( p ) cannot contain a 2 × 2-multi-grid, which implies that p is (2 , 2)-irreducible. Observe that given a point z = ( z 1 , z 2 ) ∈ C 2 and a line l : x + ay + b = 0 with non-zero slope, we have z ∈ l if and only if p ( z 1 , z 2 , a , b ) = 0. Thus, using the main theorem, the number of incidences between points in P and lines in L with a non-zero slope is bounded by O ( | P | 2 / 3+ ε | L | 2 / 3+ ε + | P | + | L | ) . Note that there are at most | P | incidences between points in P and lines in L with a zero slope, so the above bound works in general.

Introduction and the Main Theorem Applications The Algorithm Unit Distance Problem Erd˝ os’s Unit Distance Problem Given a finite set P of points in R 2 , what is the maximum number of pairs ( u , v ) ∈ P × P with � u − v � 2 = 1? Erd˝ os conjectured that the number of pairs of points in P with Euclidean distance 1 apart is bounded by O ( | P | 1+ ε ) for all ε > 0.

Introduction and the Main Theorem Applications The Algorithm Unit Distance Problem Erd˝ os’s Unit Distance Problem Given a finite set P of points in R 2 , what is the maximum number of pairs ( u , v ) ∈ P × P with � u − v � 2 = 1? Erd˝ os conjectured that the number of pairs of points in P with Euclidean distance 1 apart is bounded by O ( | P | 1+ ε ) for all ε > 0. Theorem (Spencer, Szemer´ edi, Trotter) Let P be a finite set of points in R 2 . Then, the number of pairs in P with Euclidean distance 1 apart is bounded by O ( | P | 4 / 3 ) .