It is shown that the invertible polynomial maps over a finite
field Fq , if looked at as bijections Fn,q −→ Fn,q , give all possible bijections in
the case q = 2, or q = p^r where p > 2. In the case q = 2^r where r > 1
it is shown that the tame subgroup of the invertible polynomial maps gives
only the even bijections, i.e. only half the bijections. As a consequence it
is shown that a set S ⊂ Fn,q can be a zero set of a coordinate if and only if
#S = q^(n−1).
2010 Mathematics Subject Classification: 14L99, 14R10, 20B27.
If F is a polynomial automorphism over a finite field Fq in dimension n, then it induces a permutation pqr(F) of (Fqr)n for every r О N*. We say that F can be “mimicked” by elements of a certain group of automorphisms G if there are gr О G such that pqr(gr) = pqr(F).
Derksen’s theorem in characteristic zero states that the tame automorphisms in dimension n і 3 are generated by the affine maps and the one map (x1+x22, x2,ј,...
An algorithm is described which computes generators of the kernel of derivations on k[X₁,...,Xₙ] up to a previously given bound. For w-homogeneous derivations it is shown that if the algorithm computes a generating set for the kernel then this set is minimal.
Download Results (CSV)