The mixed discriminant of n Laurent polynomials in n variables is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. The Cayley trick expresses the mixed discriminant as an A-discriminant. We show that the degree of the mixed discriminant is a piecewise linear function in the Plücker coordinates of a mixed Grassmannian. An explicit degree formula i...

When solving polynomial systems with homotopy continuation, the fundamental numerical linear algebra computations become inaccurate when two paths are in close proximity. The current best defense against this ill-conditioning is the use of adaptive precision. While sufficiently high precision indeed overcomes any such loss of accuracy, high precision can be very expensive. In this article, we d...

Invariant polynomial operators on Riemannian manifolds are well understood and the knowledge of full lists of them becomes an effective tool in Riemannian geometry, [Atiyah, Bott, Patodi, 73] is a very good example. The present short paper is in fact a continuation of [Slovák, 92] where the classification problem is reconsidered under very mild assumptions and still complete classification resu...

A continuation of research into modeling airway events of patients undergoing sedation is described. Sounds recorded at the trachea were recorded and separated by means of a threshold algorithm. The threshold was determined by the expectation maximization algorithm on filtered data. A comparison between the respiratory rate of the threshold algorithm and that of the direct airflow measure is do...

Abstract. By multidimensional matrix inversion, combined with an Ar extension of Jackson’s 8φ7 summation formula by Milne, a new multivariable 8φ7 summation is derived. By a polynomial argument this 8φ7 summation is transformed to another multivariable 8φ7 summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating 6φ5 summation. The latter is ...

Let G be a split real group with Weyl group W . Let E be an irreducible representation of W . Let V be the stable Lie algebra version of the coherent continuation representation of W . The main result of this paper is a formula for the multiplicity of E in V . The formula involves the position of E in Lusztig’s set ∐ M(G). The paper treats all quasi-split groups G as well.

A new algorithm for real root isolation of polynomial equations based on hybrid computation is presented in this paper. Firstly, the approximate (complex) zeros of the given polynomial equations are obtained via homotopy continuation method. Then, for each approximate zero, an initial box relying on the Kantorovich theorem is constructed, which contains the corresponding accurate zero. Finally,...

Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials with constant coefficients. These algorithms rely on homotopy continuation, monodromy loops, and Cauchy integrals. Several examples are presented which are c...

The objective is to demonstrate the making of Ada software available Python and Julia programmers using GPRbuild. GPRbuild project manager GNAT toolchain. With shared object files fully automated can be readily used in Julia. application build process PHCpack, a free open source package solve polynomial systems by homotopy continuation methods, written mainly Ada, with components C++, at github...