By Harm Derksen, Gregor Kemper
This booklet, the 1st quantity of a subseries on "Invariant concept and Algebraic Transformation Groups", presents a entire and up to date review of the algorithmic facets of invariant concept. a variety of illustrative examples and a cautious collection of proofs make the ebook obtainable to non-specialists.
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Extra info for Computational Invariant Theory (Encyclopedia of Mathematical Sciences)
K and denoted by Syz(II, ... , fk). More generally, we ask for the kernel of an R-homomorphism cp: Rk -+ RI between two free R-modules. If Ii := cp( ei) E RI, then the kernel of cp consists of all (hI"'" hk) E Rk with hIfl + ... k = O. Again Syz(II, ... 1 Computing Syzygies In order to explain an algorithm which computes syzygy modules, we have to give a brief introduction into Grabner bases of submodules of Rk. A monomial in Rk is an expression of the form tei with t a monomial in R. 1, with condition (i) replaced by tei > ei for all i and 1 i= t a monomial in R, and demanding (ii) for monomials h, t2 E Rk and s E R.
We will prove that h E K[ft, ... , ir] using induction on d. If d = 0, then hE K ~ K[ft, ... ,ir]. 1) with gi E K[V]. Without loss of generality, we may assume that gi is homogeneous of degree d - deg(ji) < d. 7(a)). 7(b), we obtain r r i=l i=l Because R(gi) E K[VjG is homogeneous of degree < d, we have by induction that R(gi) E K[ft, ... ,ir] for all i. We conclude that h E K[ft, ... ,ir]. 11. Ii G is a linearly reductive group acting regularly on an affine variety X, then K[X]G is finitely generated.
Although this example is very simple, it does not quite fit into the general setting, since usually we consider actions on affine varieties which are by definition reduced. 6. Let K be an algebraically closed field of characteristic O. Roberts found a (nonlinear) action of the additive group eGa on K7 such that the invariant ring is not finitely generated (see Roberts ). Recently, Daigle and Freudenburg found the following counterexample in dimension 5. Consider the action of eGa on K 5 defined by (J"' (a,b,x,y,z) = (a, b, x + (J"a 2, y + (J"(ax + b) + ~(J"2a3, z + (J"Y + ~(J"2(ax + b) + 1J(J"3a3).