Geometric complexity theory V: Efficient algorithms for Noether normalization

Mulmuley, Ketan

doi:10.1090/jams/864

Geometric complexity theory V: Efficient algorithms for Noether normalization
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by Ketan D. Mulmuley

J. Amer. Math. Soc. 30 (2017), 225-309

DOI: https://doi.org/10.1090/jams/864

Published electronically: June 21, 2016

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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 $V$ 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 $p \not \in [2,\lfloor m/2 \rfloor ]$.

(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.

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