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

Abstract:

Let $\mathbf {f} =(f_1,\dots ,f_m)$ be a sequence of polynomials of degree $\le d$ in $n$ variables $(m\ge n)$ over a field $F$. The zero-pattern of $\mathbf {f}$ at $u\in F^n$ is the set of those $i$ ($1\le i\le m$) for which $f_i(u)=0$. Let $Z_F(\mathbf {f})$ denote the number of zero-patterns of $\mathbf {f}$ as $u$ ranges over $F^n$. We prove that $Z_F(\mathbf {f}) \le \sum _{j=0}^n \binom {m}{j}$ for $d=1$ and \begin{equation*} Z_F(\mathbf {f})\le \binom {md}{n}\tag {$*$} \end{equation*} for $d\ge 2$. For $m\ge nd$, these bounds are optimal within a factor of $(7.25)^n$. The bound ($*$) improves the bound $(1+md)^n$ proved by J. Heintz (1983) using the dimension theory of affine varieties. Over the field of real numbers, bounds stronger than Heintz’s but slightly weaker than ($*$) follow from results of J. Milnor (1964), H. E. Warren (1968), 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 $\Omega (\sqrt {v/\log v})$ (the implied constant is absolute). This result was previously known over the reals (Pudlák–Rödl). 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, 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.