## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 45, Number 1 (2001), 41-73.

### On the degree of ideal membership proofs from uniform families of polynomials over a finite field

#### Abstract

Let $f_0, f_1, \dots, f_k$ be $n$--variable polynomials over a finite prime field $\fp$. A proof of the ideal membership $f_0 \in \langle f_1, \dots, f_k \rangle$ in {\em polynomial calculus} is a sequence of polynomials $h_1, \dots, h_t$ such that $h_t = f_0$, and such that every $h_i$ is either an $f_j$, $j \geq 1$, or obtained from $h_1, \dots, h_{i-1}$ by one of the two inference rules: $g_1$ and $g_2$ entail any $\fp$--linear combination of $g_1$, $g_2$, and $g$ entails $g \cdot g'$, for any polynomial $g'$. The degree of the proof is the maximum degree of the $h_i$'s. We give a condition on families $\{ f_{N,0}, \dots, f_{N,k_N}\}_{N<\omega}$ of $n_N$--variable polynomials of bounded degree implying that the minimum degree of polynomial calculus proofs of $f_{N,0}$ from $f_{N,1}, \dots, f_{N,k_N}$ cannot be bounded by an independent constant and, in fact, is $\Omega(\log(\log(N)))$. In particular, we obtain an $\Omega(\log(\log(N)))$ lower bound for the degrees of proofs of $1$ (so called {\em refutations}) of the $(N,m)$--system (defined in \cite{BIKPP}) formalizing a modular counting principle (where $m$ is fixed and not divisible by $p$, and the parameter $N$ is not divisible by $m$), and a similar lower bound for refutations of systems encoding that $N$ is composite (whenever $N$ is a prime). No bounds were previously known for these systems. The same method yields $\Omega(\log(N))$ lower bounds for the degree of coefficient polynomials in Nullstellensatz proofs. The method is based on a new result about a uniform way of generating all submoduli of tabloid moduli.

#### Article information

**Source**

Illinois J. Math., Volume 45, Number 1 (2001), 41-73.

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258138254

**Digital Object Identifier**

doi:10.1215/ijm/1258138254

**Mathematical Reviews number (MathSciNet)**

MR1849985

**Zentralblatt MATH identifier**

0984.03044

**Subjects**

Primary: 03F20: Complexity of proofs

Secondary: 12L12: Model theory [See also 03C60] 68Q15: Complexity classes (hierarchies, relations among complexity classes, etc.) [See also 03D15, 68Q17, 68Q19]

#### Citation

Krajíček, Jan. On the degree of ideal membership proofs from uniform families of polynomials over a finite field. Illinois J. Math. 45 (2001), no. 1, 41--73. doi:10.1215/ijm/1258138254. https://projecteuclid.org/euclid.ijm/1258138254