Prestable ideals and Sagbi bases

SAGBI Bases Under Composition

Our interest in the subject of this paper is inspired by Hong (1998), where Hoon Hong addresses the problem of the behavior of Gröbner bases under composition of polynomials. More precisely, let Θ be a set of polynomials, as many as the variables in our polynomial ring. The question then is under which conditions on these polynomials it is true that for an arbitrary Gröbner basis G (with respec...

SAGBI and SAGBI-Gröbner Bases over Principal Ideal Domains

Sagbi Bases of Cox-Nagata Rings

We degenerate Cox–Nagata rings to toric algebras by means of sagbi bases induced by configurations over the rational function field. For del Pezzo surfaces, this degeneration implies the Batyrev–Popov conjecture that these rings are presented by ideals of quadrics. For the blow-up of projective n-space at n + 3 points, sagbi bases of Cox–Nagata rings establish a link between the Verlinde formul...

Applications of SAGBI-bases in dynamics

The classical reduction techniques of bifurcation theory, Liapunov–Schmidt reduction and centre manifold reduction, are investigated where symmetry is present. The symmetry is given by the action of a finite or continuous group. The symmetry is exploited systematically by using the algebraic structure of the module of equivariant polynomial tuples. We generalize the concept of SAGBI-bases to mo...

Sagbi Bases in Rings of Multiplicative Invariants

Let k be a field and G be a finite subgroup of GLn(Z). We show that the ring of multiplicative invariants k[x±1 1 , . . . , x ±1 n ] G has a finite SAGBI basis if and only if G is generated by reflections.