### Abstract

Let f = (f_{1}, ..., f_{m}) be a sequence of polynomials of degree ≤ d in n variables (m ≥ n) over a field F. The zero-pattern of f at u ∈ F^{n} is the set of those i (1 ≤ i ≤ m) for which f_{i}(u) = 0. Let Z_{F}(f) denote the number of zero-patterns of f as u ranges over F^{n}. We prove that Z_{F}(f) ≤ ∑_{j}^{n}=0 (_{j}^{m}) for d = 1 and (1) Z_{F}(f) ≤ (_{n}^{md}) for d ≥ 2. For m ≥ nd, these bounds are optimal within a factor of (7.25)^{n}. The bound (1) improves the bound (1 + md)^{n} proved by J. Heintz [21] using the dimension theory of affine varieties. Over the field of real numbers, bounds stronger than Heintz's but slightly weaker than (1) follow from results of J. Milnor [28], H. E. Warren [37], and others; their proofs use techniques from real algebraic geometry. In contrast, our half-page proof is a simple application of the elementary "linear algebra bound". Heintz applied his bound to estimate the complexity of his quantifier elimination algorithm for algebraically closed fields. We give several additional applications. The first two establish the existence of certain combinatorial objects. Our first application, motivated by the "branching program" model in the theory of computing, asserts that over any field F, most graphs with v vertices have projective dimension Ω(√v/log v) (the implied constant is absolute). This result was previously known over the reals (Pudlák-Rödl [33]). The second application concerns a lower bound in the span program model for computing Boolean functions. The third application, motivated by a paper by N. Alon [2], gives nearly tight Ramsey bounds for matrices whose entries are defined by zero-patterns of a sequence of polynomials. We conclude the paper with a number of open problems.

Original language | English |
---|---|

Pages (from-to) | 717-735 |

Number of pages | 19 |

Journal | Journal of the American Mathematical Society |

Volume | 14 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jul 2001 |

### Fingerprint

### Keywords

- Affine varieties
- Algebraically closed fields
- Asymptotic counting
- Counting patterns
- Graph representation
- Linear algebra bound
- Polynomials
- Projective dimension of graphs
- Quantifier elimination
- Real algebraic geometry
- Sign-patterns
- Zero-patterns

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of the American Mathematical Society*,

*14*(3), 717-735. https://doi.org/10.1090/S0894-0347-01-00367-8

**On the number of zero-patterns of a sequence of polynomials.** / Rónyai, L.; Babai, László; Ganapathy, Murali K.

Research output: Contribution to journal › Article

*Journal of the American Mathematical Society*, vol. 14, no. 3, pp. 717-735. https://doi.org/10.1090/S0894-0347-01-00367-8

}

TY - JOUR

T1 - On the number of zero-patterns of a sequence of polynomials

AU - Rónyai, L.

AU - Babai, László

AU - Ganapathy, Murali K.

PY - 2001/7

Y1 - 2001/7

N2 - Let f = (f1, ..., fm) be a sequence of polynomials of degree ≤ d in n variables (m ≥ n) over a field F. The zero-pattern of f at u ∈ Fn is the set of those i (1 ≤ i ≤ m) for which fi(u) = 0. Let ZF(f) denote the number of zero-patterns of f as u ranges over Fn. We prove that ZF(f) ≤ ∑jn=0 (jm) for d = 1 and (1) ZF(f) ≤ (nmd) for d ≥ 2. For m ≥ nd, these bounds are optimal within a factor of (7.25)n. The bound (1) improves the bound (1 + md)n proved by J. Heintz [21] using the dimension theory of affine varieties. Over the field of real numbers, bounds stronger than Heintz's but slightly weaker than (1) follow from results of J. Milnor [28], H. E. Warren [37], and others; their proofs use techniques from real algebraic geometry. In contrast, our half-page proof is a simple application of the elementary "linear algebra bound". Heintz applied his bound to estimate the complexity of his quantifier elimination algorithm for algebraically closed fields. We give several additional applications. The first two establish the existence of certain combinatorial objects. Our first application, motivated by the "branching program" model in the theory of computing, asserts that over any field F, most graphs with v vertices have projective dimension Ω(√v/log v) (the implied constant is absolute). This result was previously known over the reals (Pudlák-Rödl [33]). The second application concerns a lower bound in the span program model for computing Boolean functions. The third application, motivated by a paper by N. Alon [2], gives nearly tight Ramsey bounds for matrices whose entries are defined by zero-patterns of a sequence of polynomials. We conclude the paper with a number of open problems.

AB - Let f = (f1, ..., fm) be a sequence of polynomials of degree ≤ d in n variables (m ≥ n) over a field F. The zero-pattern of f at u ∈ Fn is the set of those i (1 ≤ i ≤ m) for which fi(u) = 0. Let ZF(f) denote the number of zero-patterns of f as u ranges over Fn. We prove that ZF(f) ≤ ∑jn=0 (jm) for d = 1 and (1) ZF(f) ≤ (nmd) for d ≥ 2. For m ≥ nd, these bounds are optimal within a factor of (7.25)n. The bound (1) improves the bound (1 + md)n proved by J. Heintz [21] using the dimension theory of affine varieties. Over the field of real numbers, bounds stronger than Heintz's but slightly weaker than (1) follow from results of J. Milnor [28], H. E. Warren [37], and others; their proofs use techniques from real algebraic geometry. In contrast, our half-page proof is a simple application of the elementary "linear algebra bound". Heintz applied his bound to estimate the complexity of his quantifier elimination algorithm for algebraically closed fields. We give several additional applications. The first two establish the existence of certain combinatorial objects. Our first application, motivated by the "branching program" model in the theory of computing, asserts that over any field F, most graphs with v vertices have projective dimension Ω(√v/log v) (the implied constant is absolute). This result was previously known over the reals (Pudlák-Rödl [33]). The second application concerns a lower bound in the span program model for computing Boolean functions. The third application, motivated by a paper by N. Alon [2], gives nearly tight Ramsey bounds for matrices whose entries are defined by zero-patterns of a sequence of polynomials. We conclude the paper with a number of open problems.

KW - Affine varieties

KW - Algebraically closed fields

KW - Asymptotic counting

KW - Counting patterns

KW - Graph representation

KW - Linear algebra bound

KW - Polynomials

KW - Projective dimension of graphs

KW - Quantifier elimination

KW - Real algebraic geometry

KW - Sign-patterns

KW - Zero-patterns

UR - http://www.scopus.com/inward/record.url?scp=0035632206&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035632206&partnerID=8YFLogxK

U2 - 10.1090/S0894-0347-01-00367-8

DO - 10.1090/S0894-0347-01-00367-8

M3 - Article

VL - 14

SP - 717

EP - 735

JO - Journal of the American Mathematical Society

JF - Journal of the American Mathematical Society

SN - 0894-0347

IS - 3

ER -