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Relative Rank and Regularization

Amichai Lampert, Tamar Ziegler

Abstract

We introduce a new concept of rank - relative rank associated to a filtered collection of polynomials. When the filtration is trivial our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of relative bias. The main result of the paper is a relation between these two quantities over finite fields (as a special case we obtain a new proof of the results in arXiv:1902.09830). This relation allows us to get an accurate estimate for the number of points on an affine variety given by a collection of polynomials which is high relative rank (Lemma 3.2). The key advantage of relative rank is that it allows one to perform an efficient regularization procedure which is polynomial in the initial number of polynomials (the regularization process with Schmidt rank is far worse than tower exponential). The main result allows us to replace Schmidt rank with relative rank in many key applications in combinatorics, algebraic geometry and algebra. For example, we prove that any collection of polynomials $\mathcal{P}=(P_i)_{i=1}^c$ of degrees $\le d$ in a polynomial ring over an algebraically closed field of characteristic $>d$ is contained in an ideal $\mathcal{I}(\mathcal{Q})$, generated by a collection $\mathcal{Q}$ of polynomials of degrees $\le d$ which form a regular sequence, and $\mathcal{Q}$ is of size $\le A c^{A}$, where $A=A(d)$ is independent of the number of variables.

Relative Rank and Regularization

Abstract

We introduce a new concept of rank - relative rank associated to a filtered collection of polynomials. When the filtration is trivial our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of relative bias. The main result of the paper is a relation between these two quantities over finite fields (as a special case we obtain a new proof of the results in arXiv:1902.09830). This relation allows us to get an accurate estimate for the number of points on an affine variety given by a collection of polynomials which is high relative rank (Lemma 3.2). The key advantage of relative rank is that it allows one to perform an efficient regularization procedure which is polynomial in the initial number of polynomials (the regularization process with Schmidt rank is far worse than tower exponential). The main result allows us to replace Schmidt rank with relative rank in many key applications in combinatorics, algebraic geometry and algebra. For example, we prove that any collection of polynomials of degrees in a polynomial ring over an algebraically closed field of characteristic is contained in an ideal , generated by a collection of polynomials of degrees which form a regular sequence, and is of size , where is independent of the number of variables.

Paper Structure

This paper contains 11 sections, 39 theorems, 121 equations.

Table of Contents

  1. introduction
  2. Main theorem
  3. Multi-linear definitions and main theorem
  4. Some equidistribution lemmas
  5. Some reductions
  6. Many low rank approximations
  7. Regularization
  8. Fixing the approximation
  9. Nullstellensatz
  10. Reduction to the multi-linear case
  11. Adaptations of results involving rank to relative rank

Key Result

Theorem 1.3

Let $\mathbb F=\mathbb F_q$ be a finite field, let $0\le d < char(\mathbb F)$, and let $s>0$. There exists a constant $C=C(\mathbb F,s,d)$ such that the following holds: For any $n$, $P\in\mathbb F[x_1,\ldots,x_n]$ of degree $\le d$ with $bias(P) \ge q^{-s}$, we have $rk(P) \le C$.

Theorems & Definitions (90)

  • Definition 1.1: Schmidt Rank
  • Definition 1.2
  • Theorem 1.3: Bias implies low rank
  • Conjecture 1.4
  • Theorem 1.5: Size of fibers
  • Definition 1.6: Relative rank
  • Definition 1.7: Relative bias
  • Theorem 1.8
  • Definition 2.1
  • Theorem 2.2: Relative bias implies relative low rank
  • ...and 80 more