Given a polynomial ring P over field K , an element g ∈ and -subalgebra S of we deal with the problem saturating respect to i.e. computing Sat ( ) = [ − 1 ] ∩ . In general case describe procedure/algorithm compute set generators for which terminates if only it is finitely generated. Then consider more interesting when graded. particular, graded by positive matrix W indeterminate, show that choo...

In this paper we study the relation between nonhomogeneous and homogeneous Sagbi bases. As a consequence, we present a general principle of computing Sagbi bases of a subalgebra and its homogenized subalgebra, which is based on passing over to homogenized generators.

Let K be a field, D a finite distributive lattice and P the set of all join-irreducible elements of D. We show that if {y ∈ P | y ≥ x} is pure for any x ∈ P , then the Hibi ring RK(D) is level. Using this result and the argument of sagbi basis theory, we show that the homogeneous coordinate rings of Schubert subvarieties of Grassmannians are level. MSC:13F50, 13H10, 13A02, 14M15, 13P10

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Gröbner basis for the Plücker ideal of the G...

Let Cp denote the cyclic group of order p where p ≥ 3 is prime. We denote by V3 the indecomposable three dimensional representation of Cp over a field F of characteristic p. We compute a set of generators, in fact a SAGBI basis, for the ring of invariants F[V3 ⊕ V3]p . Our main result confirms the conjecture of Shank[15], for this example, that all modular rings of invariants of Cp are generate...

The maximal minors of a matrix indeterminates are universal Gröbner basis by theorem Bernstein, Sturmfels and Zelevinsky. On the other hand it is known that they not always Sagbi basis. By an experimental approach we discuss their behavior under varying monomial orders extensions to bases. These experiments motivated new implementation algorithm which organized in Singular script falls back on ...

We produce a new basis for the Schur and Weyl modules associated to a row-convex shape D. The basis is indexed by new class of \straight" tableaux which we introduce by weakening the usual requirements for standard tableaux. Spanning is proved via a new straightening algorithm for expanding elements of the representation into this basis. For skew shapes, this algorithm specializes to the classi...

The notion of initial ideal for an a polynomial ring appears in the theory Gröbner basis. Similarly to ideals, we can define algebra subalgebra ring, or more generally Laurent which is used SAGBI (Subalgebra Analogue Bases Ideals) finitely generated not always generated, and no general criterion finite generation known. aim this paper present new class subalgebras having non-finitely algebras. ...

Let W be a finite-dimensional Z/p-module over a field, k, of characteristic p. The maximum degree of an indecomposable element of the algebra of invariants, k[W ]Z/p, is called the Noether number of the representation, and is denoted by β(W ). A lower bound for β(W ) is derived, and it is shown that if U is a Z/p submodule of W , then β(U) 6 β(W ). A set of generators, in fact a SAGBI basis, is...