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 $SL_m$, 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.

💡 Research Summary

The paper “Geometric Complexity Theory V: Efficient Algorithms for Noether Normalization” addresses a fundamental algorithmic problem in algebraic geometry: constructing a normalizing linear map as guaranteed by Noether’s Normalization Lemma (NNL). The authors introduce a formal notion of “explicit” varieties—families of projective varieties whose defining equations can be described by algebraic circuits computable in polynomial time, even though the ambient dimension may be exponential in the parameter n. This formalization turns the informal “explicit” used by algebraic geometers into a precise complexity‑theoretic concept.

The main contributions are fivefold. First, for any finite‑dimensional rational representation V of the group G = SLₘ with constant m over a field of characteristic zero, the categorical quotient V//G = Spec(K