Geometric complexity theory. V: Efficient algorithms for Noether normalization (Q2826783)

The authors study a basic algorithmic problem in algebraic geometry, which is called NNL, of constructing a normalizing map as per Noether's Normalization Lemma. For general explicit varieties, as formally defined in this paper, authors give a randomized polynomial-time Monte Carlo algorithm for this problem. For some interesting cases of explicit varieties, it is provided deterministic quasi-polynomial time algorithms. These may be contrasted with the standard EXPSPACE-algorithms for these problems in computational algebraic geometry. In particular, it is shown the following:NEWLINENEWLINE (1) The categorical quotient for any finite dimensional representation \(\mathbf V\) of \(\mathbf{SL}_m\), with constant \(m\), is explicit in characteristic zero.NEWLINENEWLINE (2) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in the dimension of \(\mathbf V\).NEWLINENEWLINE (3) The categorical quotient of the space of \(r\)-tuples of \(m \times m\) matrices by the simultaneous conjugation action of \(\mathbf{SL}_m\) is explicit in any characteristic.NEWLINENEWLINE (4) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in \(m\) and \(r\) in any characteristic \(p\not\in [2, \lfloor m/2 \rfloor]\).NEWLINENEWLINE (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 \(P\).