We present formulas for computations involving companion matrix pencils as may arise in considering polynomial eigenvalue problems. In particular, we provide explicit companion matrix pencils for matrix polynomials expressed in a variety of polynomial bases including monomial, orthogonal, Newton, Lagrange, and Bernstein/Bézier bases. Additionally, we give a pair of explicit LU factors associate...

Symbolic powers of ideals are studied in the combinatorial context of monomial ideals. When the ideals are generated by quadratic squarefree monomials, the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blow-ups. As a result, perfect graphs play an important role in the theory, dual to the role played by perfect graphs in the theory of seca...

In this paper we show that there is a link between the combinatorics of the canonical basis of a quantized enveloping algebra and the monomial bases of the second author [21] arising from representations of quivers. We prove that some reparametrization functions of the canonical basis, arising from the link between Lusztig’s approach to the canonical basis and the string parametrization of the ...

Symbolic powers of ideals are studied in the combinatorial context of monomial ideals. When the ideals are generated by quadratic squarefree monomials, the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blow-ups. As a result, perfect graphs play an important role in the theory, dual to the role played by perfect graphs in the theory of seca...

The theory of “subalgebra basis” analogous to standard basis (the generalization of Gröbner bases to monomial ordering which are not necessarily well ordering [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 Ana...

The GVW algorithm, presented by Gao et al., is a signature-based algorithm for computing Gröbner bases. In this paper, a variant of GVW is presented. This new algorithm is called a monomial-oriented GVW algorithm or mo-GVW algorithm for short. The moGVW algorithm presents a new frame of GVW and regards labeled monomials instead of labeled polynomials as basic elements of the algorithm. Being di...

We show that solution sets of systems tropical differential equations can be characterised in terms monomial-freeness an initial ideal. discuss a candidate definition basis and give nonexistence result for such bases example.

In this paper, we consider a monomial ideal J / P := A[x1, . . . , xn], over a commutative ring A, and we face the problem of the characterization for the familyMf (J) of all homogeneous ideals I / P such that the A-module P/I is free with basis given by the set of terms in the Gröbner escalier N(J) of J. This family is in general wider than that of the ideals having J as initial ideal w.r.t. a...

Tensor, Clifford and Grassmann algebras belong to a wide class of non-commutative algebras that have a Poincaré-Birkhoff-Witt (PBW) “monomial” basis. The necessary and sufficient condition for an algebra to have the PBW basis has been established by T. Mora and then V. Levandovskyy as the so called “non-degeneracy condition”. This has led V. Levandovskyy to a re-discovery of the so called G-alg...