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This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construction uses a single linear differential form defined from the Jacobian matrix of the input, and defines the deflated system by applying this differential form to the original system. The advantages of this new deflation is that it does not introduce new variables and the increase in the number of equations is linear in each iteration instead of the quadratic increase of previous methods. The second construction gives the coefficients of the so-called inverse system or dual basis, which defines the multiplicity structure at the singular root. We present a system of equations in the original variables plus a relatively small number of new variables that completely deflates the root in one step. We show that the isolated simple solutions of this new system correspond to roots of the original system with given multiplicity structure up to a given order. Both constructions are “exact” in that they permit one to treat all conjugate roots simultaneously and can be used in certification procedures for singular roots and their multiplicity structure with respect to an exact rational polynomial system. [ABSTRACT FROM AUTHOR]

We give a new theoretical tool to solve sparse systems with finitely many solutions. It is based on toric varieties and basic linear algebra; eigenvalues, eigenvectors and coefficient matrices. We adapt Eigenvalue theorem and Eigenvector theorem to work with a canonical rectangular matrix (the first Koszul map) and prove that these new theorems serve to solve overdetermined sparse systems and to count the expected number of solutions. [ABSTRACT FROM AUTHOR]

Abstract: An efficient construction of two non-classical families of orthogonal polynomials is presented in the paper. The so-called half-range Chebyshev polynomials of the first and second kinds were first introduced by Huybrechs in Huybrechs (2010) . Some properties of these polynomials are also shown. Every integrable function can be represented as an infinite series of sines and cosines of these polynomials, the so-called half-range Chebyshev–Fourier (HCF) series. The second part of the paper is devoted to the efficient computation of derivatives and multiplication of the truncated HCF series, where two matrices are constructed for this purpose: the differentiation and the multiplication matrix. [Copyright &y& Elsevier]

Abstract: In this paper, we present different results related to bezoutian and residue theory. We consider, in particular, the problem of computing the structure of the quotient ring by an affine complete intersection, and an algorithm to obtain it, as conjectured in [Cardinal, J.-P., 1993. Dualité et algorithmes itératifs pour la résolution de systèmes polynomiaux. Ph.D. Thesis, Univ. de Rennes]. We analyze it in detail and prove the validity of the conjecture, for a modification of the initial method. Direct applications of the results in effective algebraic geometry are given. [Copyright &y& Elsevier]

In most algorithms that use quaternion numbers, the key operation is a quaternion multiplication, of which the efficiency and accuracy obviously determine the same properties of the whole computational scheme of a filter or transform. A digit (L‐bit)‐serial quaternion multiplier based on the distributed arithmetic (DA) using the splitting of the multiplication matrix is presented. The circuit provides the facility to compute several products of quaternion components concurrently as well as to reduce the memory capacity by half in comparison with the known DA‐based multiplier, and it is well suited for field programmable gate array (FPGA)‐based fixed‐point implementations of the algorithms. Apart from a theoretical development, the experimental design results which are obtained using a Xilinx Virtex 6 FPGA are reported. [ABSTRACT FROM AUTHOR]

We present a new eigenvalue method for solving a system of Laurent polynomial equations defining a zero-dimensional reduced subscheme of a toric compactification X of (C ∖ { 0 }) n. We homogenize the input equations to obtain a homogeneous ideal I in the Cox ring of X and generalize the eigenvalue, eigenvector theorem for root finding in affine space to compute homogeneous coordinates of the solutions. Several numerical experiments show the effectiveness of the resulting method. In particular, the method outperforms existing solvers in the case of (nearly) degenerate systems with solutions on or near the torus invariant prime divisors. [ABSTRACT FROM AUTHOR]