Set Of Monomials Research Articles

Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa–Intriligator formula

We study algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quantum double Schubert polynomials S ̃ w(x,y) , which are the Lascoux–Schützenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy identity. We define also quantum Schubert polynomials S ̃ w(x) as the Gram–Schmidt orthogonalization of some set of monomials with respect to the scalar product, defined by the Grothendieck residue. Using quantum Cauchy identity, we prove that S ̃ w(x)= S ̃ w(x,y)| y=0 and as a corollary obtain a simple formula for the quantum Schubert polynomials S ̃ w(x)=∂ ww 0 (y) S ̃ w 0 (x,y)| y=0 . We also prove the higher genus analog of Vafa–Intriligator's formula for the flag manifolds and study the quantum residues generating function. We introduce the Ehresmann–Bruhat graph on the symmetric group and formulate the equivariant quantum Pieri rule.

Monomial Orderings, Rewriting Systems, and Gröbner Bases for the Commutator Ideal of a Free Algebra

In this paper we consider a free associative algebra on three generators over an arbitrary fieldK. Given a term ordering on the commutative polynomial ring on three variables overK, we construct uncountably many liftings of this term ordering to a monomial ordering on the free associative algebra. These monomial orderings are total well orderings on the set of monomials, resulting in a set of normal forms. Then we show that the commutator ideal has an infinite reduced Gröbner basis with respect to these monomial orderings, and all initial ideals are distinct. Hence, the commutator ideal has at least uncountably many distinct reduced Gröbner bases. A Gröbner basis of the commutator ideal corresponds to a complete rewriting system for the free commutative monoid on three generators; our result also shows that this monoid has at least uncountably many distinct minimal complete rewriting systems.The monomial orderings we use are not compatible with multiplication, but are sufficient to solve the ideal membership problem for a specific ideal, in this case the commutator ideal. We propose that it is fruitful to consider such, more general, monomial orderings in non-commutative Gröbner basis theory.

Toric rings generated by special stable sets of monomials

AbstractIn this paper we consider some subalgebras of the d‐th Veronese subring of a polynomial ring, generated by stable subsets of monomials. We prove that these algebras are Koszul, showing that the presentation ideals have Gröbner bases of quadrics with respect to suitable term orders. Since the initial monomials of the elements of these Gröbner bases are square‐ free, it follows by a result of STURMFELS [S, 13.15], that the algebras under consideration are normal, and thus Cohen‐Macaulay.

Chromatic polynomials and order ideals of monomials

One expansion of the chromatic polynomial π( G, x) of a graph G relies on spanning trees of a graph. In fact, for a connected graph G of order n, one can express π(G,x)=(−1) n−1x ∑ i=1 n−1 t i(1−x) i ), where t i is the number of spanning trees with external activity 0 and internal activity i. Moreover, it is known (via commutative ring theory) that t i arises as the number of monomials of degree n − i − 1 in a set of monomials closed under division. We describe here how to explicitly carry out this construction algebraically. We also apply this viewpoint to prove a new bound for the roots of chromatic polynomials.

Involutive bases of polynomial ideals

In this paper we consider an algorithmic technique more general than that proposed by Zharkov and Blinkov for the involutive analysis of polynomial ideals. It is based on a new concept of involutive monomial division which is defined for a monomial set. Such a division provides for each monomial the self-consistent separation of the whole set of variables into two disjoint subsets. They are called multiplicative and non-multiplicative. Given an admissible ordering, this separation is applied to polynomials in terms of their leading monomials. As special cases of the separation we consider those introduced by Janet, Thomas and Pommaret for the purpose of algebraic analysis of partial differential equations. Given involutive division, we define an involutive reduction and an involutive normal form. Then we introduce, in terms of the latter, the concept of involutivity for polynomial systems. We prove that an involutive system is a special, generally redundant, form of a Gröbner basis. An algorithm for construction of involutive bases is proposed. It is shown that involutive divisions satisfying certain conditions, for example, those of Janet and Thomas, provide an algorithmic construction of an involutive basis for any polynomial ideal. Some optimization in computation of involutive bases is also analyzed. In particular, we incorporate Buchberger's chain criterion to avoid unnecessary reductions. The implementation for Pommaret division has been done in Reduce.

Floating point Gröbner bases

Bracket coefficients for polynomials are introduced. These are like specific precision floating point numbers together with error terms. Working in terms of bracket coefficients, an algorithm that computes a Gröbner basis with floating point coefficients is presented, and a new criterion for determining whether a bracket coefficient is zero is proposed. Given a finite set F of polynomials with real coefficients, let Gμ be the result of the algorithm for F and a precision value μ, and G be a true Gröbner basis of F. Then, as μ approaches infinity, Gμ converges to G coefficientwise. Moreover, there is a precision M such that if μ ≥ M, then the sets of monomials with non-zero coefficients of Gμ and G are exactly the same. The practical usefulness of the algorithm is suggested by experimental results.

Counting of paths and coefficients of the Hilbert polynomial of a determinantal ideal

Let X= [X ij] 1 ⩽ i ⩽ m 1 ⩽ i ⩽ n be an m × n of matrix of indeterminates over a field K. Abhyankar defines the index of a monomial in X ij to be the largest k such that the principal diagonal of some k × k minor of X divides the given monomial. Abhyankar has given a formula for counting the set of monomials in X ij of degree v of index at most p, satisfying a certain set of index conditions. This formula gives the Hilbert polynomial of a certain generalized determinantal ideal which can be viewed as a polynomial in v with rational coefficients. We develop a combinatorial map from this set of monomials to the set of p-tuples of nonintersecting paths in the m × n rectangular lattice of points. A path from ( a, n) to ( m, b) in a rectangular m × n array is obtained by moving either left or down at each point. The point where the path turns from down to left is called its node. Using the combinatorial map, we get formulae counting sets of p-tuples of nonintersecting paths having a fixed number of nodes in a rectangular lattice. This helps us to interpret ‘coefficients’ of Hilbert polynomials of generalized determinantal ideals combinatorially. This enables us to answer questions raised by Abhyankar in the monograph ‘Enumerative Combinatorics of Young Tableaux’.

On the Ideal Theory of Graphs

We study algebras defined by finite sets G = { M 1, ..., M q } of monomials of a polynomial ring R. There are two basic algebras: (i) k[ G] = k[ M 1, ..., M q ], the k-subalgebra of R spanned by the M i , and (ii) the quotient ring R/ I( G), where I( G) = ( M 1, ..., M q ). They come together in the construction of the Rees algebra R ( I( G)) of the ideal I( G). The emphasis is almost entirely on sets of squarefree monomials of degree two and their attached graphs. The main results are assertions about the Cohen-Macaulay behaviour of the Koszul homology of I( G), and how normality or Cohen-Macaulayness of one of the algebras can be read off the properties of the graph or in the other algebra.

Finding sparse systems of parameters

For several computational procedures such as finding radicals and Noether normalizations, it is important to choose as sparse as possible a system of parameters in a polynomial ideal or modulo a polynomial ideal. We describe new strategies for these tasks, thus providing solutions to problems (1) and (2) posed by D. Eisenbud et al. [Invent. Math. 110 (1992) 207-236]. To accomplish the first task we introduce a notion of “setwise complete intersection”. We prove that a set of monomials generating an ideal of codimension c in a polynomial ring can be partitioned into c disjoint sets forming a setwise complete intersection, although the corresponding result is false for arbitrary sets of polynomials. We reduce the general case to the monomial case by a deformation argument. For homogeneous ideals the output is homogeneous. Our analysis of the second task is based on a concept of Noether complexity for homogeneous ideals and its characterization in terms of Chow forms.

Generalized coinsertion and standard multitableaux

We extend the Robinson-Schensted-Knuth insertion procedure to tableaux over totally ordered sets and establish a correspondence from the set of monomials of certain types to the set of tableaux over totally ordered sets. It turns out that for the two dimensional case the said correspondence is bijective and for dimension greater than two, it is injective and the image of the set of monomials properly contain the set of standard tableaux. As a consequence we prove that the Straightening Law of Doubilet-Rota-Stein is not valid in the case of higher dimensional matrices.

Topological Types of Quasi-Ordinary Singularities

A germ $(X,x)$ of a complex analytic hypersurface in ${\mathbb {C}^{d + 1}}$ is quasi-ordinary if it can be represented as the image of an open neighborhood of $0$ in ${\mathbb {C}^d}$ under the map $({s_1}, \ldots ,{s_d}) \mapsto (s_1^n, \ldots ,s_d^n,\zeta ({s_1}, \ldots ,{s_d})),\;n > 0$, where $\zeta$ is a convergent power series. It is shown that the topological type of the singularity $(X,x) \subset ({\mathbb {C}^{d + 1}},0)$ is determined by a certain set of fractional monomials, called the characteristic monomials, appearing in the fractional power series $\zeta (t_1^{1/n}, \ldots ,t_d^{1/n})$.

Topological types of quasi-ordinary singularities

A germ ( X , x ) (X,x) of a complex analytic hypersurface in C d + 1 {\mathbb {C}^{d + 1}} is quasi-ordinary if it can be represented as the image of an open neighborhood of 0 0 in C d {\mathbb {C}^d} under the map ( s 1 , … , s d ) ↦ ( s 1 n , … , s d n , ζ ( s 1 , … , s d ) ) , n > 0 ({s_1}, \ldots ,{s_d}) \mapsto (s_1^n, \ldots ,s_d^n,\zeta ({s_1}, \ldots ,{s_d})),\;n > 0 , where ζ \zeta is a convergent power series. It is shown that the topological type of the singularity ( X , x ) ⊂ ( C d + 1 , 0 ) (X,x) \subset ({\mathbb {C}^{d + 1}},0) is determined by a certain set of fractional monomials, called the characteristic monomials, appearing in the fractional power series ζ ( t 1 1 / n , … , t d 1 / n ) \zeta (t_1^{1/n}, \ldots ,t_d^{1/n}) .

Generalized roinsertive correspondence between multitableaux and multimonomials

By generalizing the Robinson-Schensted-Knuth insertion procedure, we establish a bijective correspondence between the set of certain types of bitableaux over totally ordered sets and the set of indexed monomials satisfying certain constraints. As a consequence we show that the Straightening Law of Doubilet-Rota-Stein is not valid in the case of ‘higher dimensional’ matrices. In greater detail: In the classical two dimensional case, the said Law says that the standard monomials in the minors of a (rectangular) matrix X, which correspond to standard bitableaux, form a vector space basis of the polynomial ring K[ X] in the indeterminate entries of X over the coefficient field K. Now we may ask what happens to this when we consider ‘higher dimensional’ matrices by using cubical, 4-way,…, q-way determinants which were already introduced by Cayley in 1843. In the present paper we show that, for q>2, the standard monomials in the multiminors of the multimatrix X do not span the polynomial ring K[ X]; in a forthcoming paper it will be shown that they are linearly independent over K.

ON A SHIRSHOV BASIS OF RELATIVELY FREE ALGEBRAS OF COMPLEXITY $ n$

A Shirshov basis is a set of elements of an algebra over which has bounded height in the sense of Shirshov.A description is given of Shirshov bases consisting of words for associative or alternative relatively free algebras over an arbitrary commutative associative ring with unity. It is proved that the set of monomials of degree at most is a Shirshov basis in a Jordan PI-algebra of degree . It is shown that under certain conditions on (satisfied by alternative and Jordan PI-algebras), if each factor of with nilpotent projections of all elements of is nilpotent, then is a Shirshov basis of if generates as an algebra.Bibliography: 12 titles.

Analysis of straightening formula

The straightening formula has been an essential part of a proof showing that the set of standard bitableaux (or the set of standard monomials in minors) gives a free basis for a polynomial ring in a matrix of indeterminates over a field. The straightening formula expresses a nonstandard bitableau as an integral linear cobmbination of standard bitableaux. In this paper we analyse the exchanges in the process of straightening a nonstandard pure tableau of depth two. We give precisely the number of steps required to straighten a given violation of a nonstandard tableau. We also characterise the violation which is eliminated in a single step.

Fully Symmetric Integration Formulas for the Surface of the Sere in S Dimensions

Methods of polynomial degree d to approximate integrals over the surface of the s-dimensional sphere are discussed. Such methods are constructed by choosing the abscissas and weights so that the formulas integrate a certain subset of the polynomials of maximum degree d, exactly. A simple set of monomials is described, the exact integration of which will ensure that the method has the required polynomial degree. Results are obtained which enable the derivation of consistency conditions on the rule structures. An example of a new s-dimensional degree-11 rule is given.

Translation invariant subspaces of finite dimension

Abstract : Several simple proofs are given of the following theorem. Every finite dimensional translation invariant subspace W of the continuous functions on (minus infinity, infinity), or of the Schwartz distributions on (minus infinity, plus infinity), is spanned by a set of exponential monomials t to the (mu-1) power times e to the (lambda t) power, lambda lambda sub 1, ..., lambda sub k. Thus, W is the solution space o a homogeneous linear ordinary differential equation with constant coefficients. (Author)