Geometric Complexity Theory V: Efficient algorithms for Noether Normalization

TL;DR

This paper develops efficient randomized and deterministic algorithms for Noether Normalization, a fundamental problem in algebraic geometry, achieving significant improvements over traditional exponential-space methods, with implications for geometric complexity theory.

Contribution

It introduces polynomial-time randomized and quasi-polynomial deterministic algorithms for Noether Normalization on explicit varieties, advancing computational algebraic geometry.

Findings

Deterministic quasi-polynomial algorithms for categorical quotients of $SL_m$ representations.

Explicit algorithms for NNL in characteristic zero and large enough characteristic.

Connections to geometric complexity theory and the permanent hypothesis.

Abstract

We study a basic algorithmic problem in algebraic geometry, which we call NNL, of constructing a normalizing map as per Noether's Normalization Lemma. For general explicit varieties, as formally defined in this paper, we give a randomized polynomial-time Monte Carlo algorithm for this problem. For some interesting cases of explicit varieties, we give deterministic quasi-polynomial time algorithms. These may be contrasted with the standard EXPSPACE-algorithms for these problems in computational algebraic geometry. In particular, we show that: (1) The categorical quotient for any finite dimensional representation $V$ of $SL_m$, with constant $m$, is explicit in characteristic zero. (2) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in the dimension of $V$. (3) The categorical quotient of the space of $r$-tuples of $m \times m$ matrices by the simultaneous conjugation action of $SL_m$ is explicit in any characteristic. (4) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in $m$ and $r$ in any characteristic $p$ not in $[2,\ m/2]$. (5) NNL for every explicit variety in zero or large enough characteristic can be solved deterministically in quasi-polynomial time, assuming the hardness hypothesis for the permanent in geometric complexity theory. The last result leads to a geometric complexity theory approach to put NNL for every explicit variety in P.