Geometric Complexity Theory V: Efficient algorithms for Noether Normalization

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 SLm, with constant m, is explicit in characteristic zero. (2) NNL for this categorical quotient can be solved deterministically in time quasipolynomial in the dimension of V . (3) The categorical quotient of the space of r-tuples ofm×mmatrices by the simultaneous conjugation action of SLm is explicit in any characteristic. (4) NNL for this categorical quotient can be solved deterministically in time quasipolynomial in m and r in any characteristic p 6∈ [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.