Related papers: Resultant-based Elimination in Ore Algebra
Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…
Resultants and Gr\"obner bases are crucial tools in studying polynomial elimination theory. We investigate relations between the variety of the resultant of two polynomials and the variety of the ideal they generate. Then we focus on the…
We are concerned with the problem of decomposing the parameter space of a parametric system of polynomial equations, and possibly some polynomial inequality constraints, with respect to the number of real solutions that the system attains.…
The resultant of two univariate polynomials is an invariant of great importance in commutative algebra and vastly used in computer algebra systems. Here we present an algorithm to compute it over Artinian principal rings with a modified…
We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of…
We study the complexity of computing the real solutions of a bivariate polynomial system using the recently proposed algorithm BISOLVE. BISOLVE is a classical elimination method which first projects the solutions of a system onto the $x$-…
Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders.…
Differential resultant formulas are defined, for a system $\mathcal{P}$ of $n$ ordinary Laurent differential polynomials in $n-1$ differential variables. These are determinants of coefficient matrices of an extended system of polynomials…
Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula…
This paper gives an explicit method for computing the resultant of any sparse unmixed bivariate system with given support. We construct square matrices whose determinant is exactly the resultant. The matrices constructed are of hybrid…
In this paper, a new triangular decomposition algorithm is proposed for ordinary differential polynomial systems, which has triple exponential computational complexity. The key idea is to eliminate one algebraic variable from a set of…
Differential-elimination algorithms apply a finite number of differentiations and eliminations to systems of partial differential equations. For systems that are polynomially nonlinear with rational number coefficients, they guarantee the…
The linear complete differential resultant of a finite set of linear ordinary differential polynomials is defined. We study the computation by linear complete differential resultants of the implicit equation of a system of $n$ linear…
We propose an algorithm for determining the irreducible polynomials over finite fields, based on the use of the companion matrix of polynomials and the generalized Jordan normal form of square matrices.
A fundamental problem in computational algebraic geometry is the computation of the resultant. A central question is when and how to compute it as the determinant of a matrix. whose elements are the coefficients of the input polynomials…
We apply the bi-moment determinant method to compute a representation of the matrix product algebra -- a quadratic algebra satisfied by the operators $\mathbf{d}$ and $\mathbf{e}$ -- for the five parameter ($\alpha$, $\beta$, $\gamma$,…
In this note, we consider the resultant of systems of homogeneous multivariate polynomials which are equivariant under the action of direct product of two symmetric groups. We establish a decomposition formula for the resultant of such…
In this paper, the result of applying iterative univariate resultant constructions to multivariate polynomials is analyzed. We consider the input polynomials as generic polynomials of a given degree and exhibit explicit decompositions into…
The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the…
In this paper we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial…