Invariant algebraic sets and symmetrization of polynomial systems
Consider the variety \(V \subset \mathbb{C}^n\) of a finite set of polynomials \(F \subset \mathbb{C}[z_1,\ldots ,z_n]\). Then one observes that \(V\) may be invariant under the action of a group \(G\) independent of the polynomial set \(F\). The aim of this paper is to construct a set of invariants that define the same variety \(V\), but outside of some proper subvariety of \(V\). For this, the considered group actions as well as the notion of section of degree \(e\) to the orbits are defined. It has been shown that the coefficients of a reduced Gröbner basis of the orbit-section ideal is a finite set of generating rational invariants. Then, one is able to give a constructive definition of symmetrization w.r.t. a given section to the orbits. Now a set of invariant functions \(\tilde{F}\) (polynomials in the constructed set of generating rational invariants) can be constructed such that the zero set of \(F\) is the same as that of \(\tilde{F}\), except outside of some proper subvariety of \(V\).
scientific article; zbMATH DE number 7074701
scientific article