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.