We study Bezier surfaces defined over triangular domains. We have seen how spaces of bivariate polynomials can be constructed as tensor-products of the univariate ones. An alternative choice of polynomial space is, for each n ≥ 0, the space of polynomials of the form p(x, y) = 0≤i+j≤n a i,j x i y j , where we understand that i ≥ 0 and j ≥ 0 in the summation. As in the univariate case, we denote...

We give a criterion for checking the Cohen-Macaulayness of the tangent cone of a monomial curve by using the Gröbner basis. For a family of monomial curves, we give the full description of the defining ideal of the curve and its tangent cone at the origin. By using this family of curves and their extended versions to higher dimensions, we prove that the minimal number of generators of a Cohen-M...

A major area in neuroscience is the study of how the brain processes spatial information. Neurons in the brain represent external stimuli via neural codes. These codes often arise from regions of space called receptive fields: each neuron fires at a high rate precisely when the animal is in the corresponding receptive field. Much research in this area has focused on understanding what features ...

Let U − q (g) be the negative part of the quantized universal enveloping algebra constructed from a Cartan matrix associated to a complex semisimple Lie algebra g. Let λ be a dominant integral weight and V (λ) the irreducible U − q (g)-module with highest weight λ. There is on the one hand the canonical basis for U − q (g) [4, 12, 13] and on the other the standard monomial theoretic basis for t...

For any set representation (permutation representation) of the symmetric group Sn, we give combinatorial interpretation for coefficients of its Frobenius character expanded in the basis of monomial symmetric functions.

Ezra Miller and Bernd Sturmfels Department of Mathemati s, University of California, Berkeley CA 94720, USA enmiller math.berkeley.edu, bernd math.berkeley.edu Abstra t. Gr obner basis theory redu es questions about systems of polynomial equations to the ombinatorial study of monomial ideals, or stair ases. This arti le gives an elementary introdu tion to urrent resear h in this area. After re...

We call superpartitions the indices of the eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model. We obtain an ordering on superpartitions from the explicit action of the model’s Hamiltonian on monomial superfunctions. This allows to define Jack superpolynomials as the unique eigenfunctions of the model that decompose triangularly, with respect to t...

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...

Recently analog circuit designers are interested in structured optimization techniques to automate the process of CMOS circuit design. Geometric programming, which makes use of monomial and posynomial expressions to model MOSFET parameters, represents one such approach. The extent of accuracy in finding a global optimal solution using this approach depends on the formulation of circuit and devi...