SAGBI Basis Research Articles

Powers of generalized binomial edge ideals of path graphs

In this paper, we study the powers of the generalized binomial edge ideal [Formula: see text] of a path graph [Formula: see text]. We explicitly compute their regularities and determine the limit of their depths. We also show that these ordinary powers coincide with their symbolic powers. Additionally, we study the Rees algebra and the special fiber ring of [Formula: see text] utilizing Sagbi basis theory. In particular, we obtain precise formulas for the regularity of these blowup algebras.

Modular invariants of a vector and a covector for some elementary abelian p-groups

Let F p be the prime field of order p > 0 and G be an elementary abelian p-group. For some n-dimensional cohyperplane G-representations V over F p , we show that F p [ V ⊕ V * ] G , the invariant ring of a vector and a covector is a complete intersection by exhibiting an explicit generating set (in fact, a SAGBI basis) and exposing all relations among the generators.

Rees algebras of unit interval determinantal facet ideals

Using SAGBI basis techniques, we find Gröbner bases for the presentation ideals of the Rees algebras and special fiber rings of unit interval determinantal facet ideals. In particular, we show that unit interval determinantal facet ideals are of fiber type and that their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains and have rational singularities.

Sagbi combinatorics of maximal minors and a Sagbi algorithm

The maximal minors of a matrix of indeterminates are a universal Gröbner basis by a theorem of Bernstein, Sturmfels and Zelevinsky. On the other hand it is known that they are not always a universal Sagbi basis. By an experimental approach we discuss their behavior under varying monomial orders and their extensions to Sagbi bases. These experiments motivated a new implementation of the Sagbi algorithm which is organized in a Singular script and falls back on Normaliz for the combinatorial computations. In comparison to packages in the current standard distributions of Macaulay 2, version 1.21, and Singular, version 4.2.1 and a package intended for CoCoA 5.4.2, it extends the range of computability by at least one order of magnitude.

Modular computations of standard bases for subalgebras

We develop a theory, algorithms and implementations to compute sagbi bases for local orderings in a parallel and modular setting. By this method, we can calculate sagbi basis with large size coefficients efficiently by avoiding intermediate coefficients growth. We have implemented these algorithms including modular and parallel implementations and give timings. We also discuss the verification of the modular algorithm to compute sagbi bases.

Rees algebra and special fiber ring of binomial edge ideals of closed graphs

In this paper, I compute the regularity of the Rees algebra of binomial edge ideals of closed graphs. I obtain a lower bound for the regularity of the Rees algebra of binomial edge ideals. I also present some algebraic properties of the Rees algebra and special fiber ring of binomial edge ideals of closed graphs via algebraic properties of their initial algebra and Sagbi basis theory. I obtain an upper bound for the regularity of the special fiber ring of binomial edge ideals of closed graphs.

Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux

We study Gröbner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by matching field tableaux in the sense of Sturmfels and Zelevinsky in [31]. We prove that these ideals are all quadratically generated and they yield a SAGBI basis of the Plücker algebra. This leads to a new family of toric degenerations of Grassmannians. Moreover, we apply our results to construct a family of Gröbner degenerations of Schubert varieties inside Grassmannians. We provide a complete characterization of toric ideals among these degenerations in terms of the combinatorics of matching fields, permutations and semi-standard tableaux.

Homogeneous Sagbi Bases Under Polynomial Composition

The process of replacing indeterminates in a Polynomial with other polynomials is the polynomial composition. Homogeneous Sagbi bases are the Sagbi bases generated by the subset of homogeneous polynomials. In this article we present adequate and essential criterion on a set of polynomials to guarantee that the composed set \(S\circ \ominus\) is Homogeneous Sagbi basis whenever \(S\) is a Homogeneous Sagbi basis.

A new algorithm for computing SAGBI bases up to an arbitrary degree

We present a new algorithm for computing a SAGBI basis up to an arbitrary degree for a subalgebra generated by a set of homogeneous polynomials. Our idea is based on linear algebra methods which cause a low level of complexity and computational cost. We then use it to solve the membership problem in subalgebras.

FURTHER ON THE COMPOSITION OF SAGBI BASES

Composition is the operation of replacing variables in a polynomial by other polynomials. In this paper, we show that composition commutes with SAGBI basis computation (possibly under different monomial orderings) if the leading monomials of the composition polynomials are a permuted powering.

The invariant rings of the Sylow groups of GU(3,q2), GU(4,q2), Sp(4,q) and O+(4,q) in the natural characteristic

Let G be a Sylow p-subgroup of the unitary groups GU(3,q2), GU(4,q2), the symplectic group Sp(4,q) and, for q odd, the orthogonal group O+(4,q). In this paper we construct a presentation for the invariant ring of G acting on the natural module. In particular we prove that these rings are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant form defining the corresponding classical group. We also show that these generators form a SAGBI basis and the invariant ring for G is a complete intersection.

Subalgebra Analogue to H-Basis for Ideals

The H-basis concept allows an investigation of multivariate polynomial spaces degree by degree. In this paper we present the analogue of Hbases for subalgebras in polynomial rings, we call them SH-bases. We present their connection to the Sagbi basis concept, characterize SH-basis and show how to construct them.

Converting subalgebra bases with the Sagbi walk

We present an algorithm which converts a given Sagbi basis of a polynomial K-subalgebra A with respect to one monomial ordering to the Sagbi basis of A with respect to another monomial ordering, under the assumption that the subalgebra A admits a finite Sagbi basis with respect to all monomial orderings. The Sagbi walk method converts a Sagbi basis by partitioning the computations following a path in the Sagbi fan.

Subalgebra Analogue to standard basis for ideal

A theory of “subalgebra basis” analogous to standard basis (the generalization of Gröbner bases to monomial orderings which are not necessarily well orderings [1]) for ideals in polynomial rings over a field is developed. We call these bases “SASBI Basis” for “Subalgebra Analogue to Standard Basis for Ideals”. The case of global orderings, here they are called “SAGBI Basis” for “Subalgebra Analogue to Gröbner Basis for Ideals”, is treated in [6]. Sasbi bases may be infinite. In this paper we consider subalgebras admitting a finite Sasbi basis and give algorithms to compute them.

The invariants of the third symmetric power representation of [formula omitted

For a prime p > 3 , we compute a finite generating set for the SL 2 ( F p ) -invariants of the third symmetric power representation. The proof relies on the construction of an infinite SAGBI basis and uses the Hilbert series calculation of Hughes and Kemper.

SAGBI bases for the Kernel of certain locally nilpotent K-derivations on polynomial rings.

In this paper is obtained an algebraic-computational property related to the theory of SAGBI bases for the specific case of the kernel of some locally nilpotent derivations on the polynomial ring in several variables over a eld of characteristic zero. The result achieved via explicit calculation is an algebraic characterization and a nite SAGBI basis for the kernel of a certain class of locally nilpotent derivations.

Vector invariants for the two-dimensional modular representation of a cyclic group of prime order

In this paper, we study the vector invariants of the 2-dimensional indecomposable representation V 2 of the cyclic group, C p , of order p over a field F of characteristic p, F [ m V 2 ] C p . This ring of invariants was first studied by David Richman (1990) [20] who showed that the ring required a generator of degree m ( p − 1 ) , thus demonstrating that the result of Noether in characteristic 0 (that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group) does not extend to the modular case. He also conjectured that a certain set of invariants was a generating set with a proof in the case p = 2 . This conjecture was proved by Campbell and Hughes (1997) in [3]. Later, Shank and Wehlau (2002) in [24] determined which elements in Richman's generating set were redundant thereby producing a minimal generating set. We give a new proof of the result of Campbell and Hughes, Shank and Wehlau giving a minimal algebra generating set for the ring of invariants F [ m V 2 ] C p . In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for F [ m V 2 ] C p . Further, our results provide a procedure for finding an explicit decomposition of F [ m V 2 ] into a direct sum of indecomposable C p -modules. Finally, noting that our representation of C p on V 2 is as the p-Sylow subgroup of SL 2 ( F p ) , we describe a generating set for the ring of invariants F [ m V 2 ] SL 2 ( F p ) and show that ( p + m − 2 ) ( p − 1 ) is an upper bound for the Noether number, for m > 2 .

A SAGBI Basis For 𝔽[V2 ⊕ V2 ⊕ V3]Cp

AbstractLet Cp denote the cyclic group of order p, where p ≥ 3 is prime. We denote by Vn the indecomposable n dimensional representation of Cp over a field of characteristic p. We compute a set of generators, in fact a SAGBI basis, for the ring of invariants [V2 ⊕ V2 ⊕ V3]Cp .

SAGBI bases for rings of invariant Laurent polynomials

Let k be a field, let L n = k[x ±1 1 , x ±1 n ] be the Laurent polynomial ring in n variables and let G be a finite group of k-algebra automorphisms of L n . We give a necessary and sufficient condition for the ring of invariants L G n to have a SAGBI basis. We show that if this condition is satisfied, then L G n has a SAGBI basis relative to any choice of coordinates in L n and any term order.

A Sufficient Condition for a Hibi Ring to Be Level and Levelness of Schubert Cycles

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 ℛ K (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.