SAGBI Bases Research Articles

Sagbi bases over the product of rings

Let $R$ be a commutative ring such that $R=R_1\times \cdots \times R_n$. In this paper, we give a method to compute (strong) Sagbi bases for subalgebras of a polynomial ring over $R$ provided that these bases are computable in a polynomial ring over $R_i$ for $1\leq i \leq n$. As an application, we prove the existence of strong Sagbi bases for subalgebras in a polynomial ring over a principal ideal ring.

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.

Subalgebras in K[x] of small codimension

We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in mathbb {K}[x]. The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type f(alpha )=f(beta ) for alpha ,beta in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp(A) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras.

Toric degenerations of flag varieties from matching field tableaux

We present families of tableaux which interpolate between the classical semi-standard Young tableaux and matching field tableaux. Algebraically, this corresponds to SAGBI bases of Plücker algebras . We show that each such family of tableaux leads to a toric ideal, that can be realized as initial of the Plücker ideal, hence a toric degeneration for the flag variety.

Saturations of subalgebras, SAGBI bases, and U-invariants

Given a polynomial ring P over a field K, an element g∈P, and a K-subalgebra S of P, we deal with the problem of saturating S with respect to g, i.e. computing Satg(S)=S[g,g−1]∩P. In the general case we describe a procedure/algorithm to compute a set of generators for Satg(S) which terminates if and only if it is finitely generated. Then we consider the more interesting case when S is graded. In particular, if S is graded by a positive matrix W and g is an indeterminate, we show that if we choose a term ordering σ of g-DegRev type compatible with W, then the two operations of computing a σ-SAGBI basis of S and saturating S with respect to g commute. This fact opens the doors to nice algorithms for the computation of Satg(S). In particular, under special assumptions on the grading one can use the truncation of a σ-SAGBI basis and get the desired result. Notably, this technique can be applied to the problem of directly computing some U-invariants, classically called semi-invariants, even in the case that K is not the field of complex numbers.

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.

Crystal bases and Newton–Okounkov bodies

Let G be a connected reductive algebraic group. We prove that the string parametrization of a crystal basis for a finite dimensional irreducible representation of G extends to a natural valuation on the field of rational functions on the flag variety G/B, which is a highest term valuation corresponding to a coordinate system on a Bott-Samelson variety. This shows that the string polytopes associated to irreducible representations, can be realized as Newton-Okounkov bodies for the flag variety. This is closely related to an earlier result of A. Okounkov for the Gelfand-Cetlin polytopes of the symplectic group. As a corollary we recover a multiplicativity property of the canonical basis due to P. Caldero. We generalize the results to spherical varieties. From these the existence of SAGBI bases for the homogeneous coordinate rings of flag and spherical varieties, as well as their toric degenerations follow, recovering previous results of Caldero, Alexeev-Brion and the author.

On the (De)homogenization of Sagbi Bases

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

Weitzenböck derivations of nilpotency 3

Abstract We consider a Weitzenböck derivation Δ acting on a polynomial ring R = K [ ξ 1 , ξ 2 , ... , ξ m ] $ R=K[\xi _1,\xi _2,\ldots ,\xi _m] $ over a field K of characteristic 0. The K-algebra R Δ = { h ∈ R ∣ Δ ( h ) = 0 } ${R^\Delta = \lbrace h \in R \mid \Delta (h) = 0\rbrace }$ is called the algebra of constants. Nowicki considered the case where the Jordan matrix for Δ acting on R 1, the degree 1 component of R, has only Jordan blocks of size 2. He conjectured that a certain set generates R Δ $R^{\Delta }$ in that case. Recently Khoury, Drensky and Makar-Limanov and Kuroda have given proofs of Nowicki's conjecture. Here we consider the case where the Jordan matrix for Δ acting on R 1 has only Jordan blocks of size at most 3. We use combinatorial methods to give a minimal set of generators 𝒢 for the algebra of constants R Δ $R^{\Delta }$ . Moreover, we show how our proof yields an algorithm to express any h ∈ R Δ $h \in R^\Delta $ as a polynomial in the elements of 𝒢. In particular, our solution shows how the classical techniques of polarization and restitution may be used to augment the techniques of SAGBI bases to construct generating sets for subalgebras.

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.

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 formula and phylogenetic algebraic geometry, and we use this to answer questions due to D’Cruz–Iarrobino and Buczyńska–Wiśniewski. Inspired by the zonotopal algebras of Holtz and n s Ron, our study emphasizes explicit computations, and offers a new approach to Hilbert functions of fat points.

Using symmetries in the eigenvalue method for polynomial systems

One way of solving polynomial systems of equations is by computing a Gröbner basis, setting up an eigenvalue problem and then computing the eigenvalues numerically. This so-called eigenvalue method is an excellent bridge between symbolic and numeric computation, enabling the solution of larger systems than with purely symbolic methods. We investigate the case that the system of polynomial equations has symmetries. For systems with symmetry, some matrices in the eigenvalue method turn out to have special structure. The exploitation of this special structure is the aim of this paper. For theoretical development we make use of SAGBI bases of invariant rings. Examples from applications illustrate our new approach.

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.

Standard monomial theory for flag algebras of [formula omitted] and [formula omitted

Let G be the general linear group or the symplectic group over the complex numbers, and U be its maximal unipotent subgroup. We study standard monomial theory for the ring of regular functions on G / U , called the flag algebra, using the philosophy of Gröbner bases and SAGBI bases combined with classical invariant theory. From the realization of the flag algebra in a concrete polynomial setting, we obtain explicit standard monomial bases for irreducible representations. We also recreate known combinatorics of Young tableaux and Gelfand–Tsetlin patterns, and toric degenerations of flag varieties from the structure of leading monomials.

Using resultants for SAGBI basis verification in the univariate polynomial ring

A resultant-type identity for univariate polynomials is proved and used to characterise SAGBI bases of subalgebras generated by two polynomials. A new equivalent condition, expressed in terms of the degree of a field extension, for a pair of univariate polynomials to form a SAGBI basis is derived.

SAGBI bases and degeneration of spherical varieties to toric varieties

Let X \subset Proj(V) be a projective spherical G-variety, where V is a finite dimensional G-module and G = SP(2n, C). In this paper, we show that X can be deformed, by a flat deformation, to the toric variety corresponding to a convex polytope \Delta(X). The polytope \Delta(X) is the polytope fibred over the moment polytope of X with the Gelfand-Cetlin polytopes as fibres. We prove this by showing that if X is a horospherical variety, e.g. flag varieties and Grassmanians, the homogeneous coordinate ring of X can be embedded in a Laurent polynomial algebra and has a SAGBI basis with respect to a natural term order. Moreover, we show that the semi-group of initial terms, after a linear change of variables, is the semi-group of integral points in the cone over the polytope \Delta(X). The results of this paper are true for other classical groups, provided that a result of A. Okounkov on the representation theory of SP(2n,C) is shown to hold for other classical groups.

Prestable ideals and Sagbi bases

In order to find a reasonable class of squarefree monomial ideals $I$ for which the toric ideal of the Rees algebra of $I$ has a quadratic Gröbner basis, the concept of prestable ideals will be introduced. Prestable ideals arising from finite pure posets together with their application to Sagbi bases will be discussed.

A finite universal SAGBI basis for the kernel of a derivation

To find and to calculate generating sets for invariant rings i s a fundamental prob- lem in invariant theory with a long tradition. With the progress of computers, the sig- nificance of computational methods in this field has increase d. The SAGBI bases are the sets of generators of a subalgebra of a polynomial ring which have certain com- putational property. These are the natural Subalgebra Analogue to Grobner Bases for Ideals introduced at the end of 1980's by Robbiano and Sweed ler (20) and Kapur and Madlener (8), independently. There are indeed some applications of the SAGBI bases to invariant theory. The algorithm of Stillman and Tsai (23) gives a method for computing generating sets for certain invariant rings by using this notion. However, compared with the theory of Grobner bases, that of SAGBI bases has made a slow progress, and many basic problems remaining unsolved. The purpose of this paper is to investigate the properties of a SAGBI basis for the kernel of a derivation on a poly- nomial ring. The kernel of a derivation on a polynomial ring is closely related to an invari- ant ring. It is an important object in the study of invariant t heory and the fourteenth problem of Hilbert. It is well-known that some kind of derivation corresponds to an action of one-dimensional additive group, and the kernel and the invariant subring are the same. Moreover, various counterexamples to the fourteenth problem of Hilbert can be described as the kernel of a derivation. Nagata's counter example (17) and Roberts' counterexample (22) were described as this by Derksen (2) and by Deveney and Fin- ston (4), respectively. Nowicki showed that the invariant subring for a linear action of a connected linear algebraic group on a polynomial ring is obtained as the kernel of a derivation (18). Recently, new counterexamples to the fourteenth problem of Hilbert were constructed by using the kernel of a derivation by several people (cf. (1), (6), (10), (13)). We believe that a computational methods will give us further progress in this field. In this paper, is always a field of characteristic zero except Section 6. Let