Geometric complexity theory V: Efficient algorithms for Noether normalization

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 the following: (1) The categorical quotient for any finite dimensional representation V V of S L m SL_m , with constant m 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 V . (3) The categorical quotient of the space of r r -tuples of m × m m \times m matrices by the simultaneous conjugation action of S L m SL_m is explicit in any characteristic. (4) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in m m and r r in any characteristic p ∉ [ 2 , ⌊ m / 2 ⌋ ] p \not \in [2,\lfloor m/2 \rfloor ] . (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.