Comprehensive Gröbner bases for parametric polynomial ideals were introduced, constructed, and studied by the author in 1992. Since then the construction has been implemented in the computer algebra systems ALDES/SAC-2, MAS, REDUCE and MAPLE. A ...

In this paper we consider the problem of the use of approximate arithmetics in Gröbner basis computation. This is useful to reduce the cost of integer arithmetic, but is especially necessary for overdetermined systems whose coefficients are only ...

We give an alternative definition of comprehensive Gröbner bases in terms of Gröbner bases in polynomial rings over commutative Von Neumann regular rings. Our comprehensive Gröbner bases are defined as Gröbner bases in polynomial rings over certain ...

The well-known technique of adic-lifting for linear-system solution is studied. Some new methods are developed and applied to get algorithms for the following problems over the ring of univariate polynomials with coefficients from a field: rational ...

We study the representation of the solutions of a polynomial system by triangular sets, and concentrate on the positive-dimensional case. We reduce to dimension zero by placing the free variables in the base-field, so the solutions can be represented by ...

Given a rational parametrization of an algebraic surface, we try to reduce the degree by a suitable reparametrization. We give an algorithm that produces a parametrization with a degree that is at most twice the minimal degree. The problem is closely ...

This paper presents a new encoding scheme for real algebraic number manipulations which enhances current Axiom's Real Closure. Algebraic manipulations are performed using different instantiations of sub-resultant like algorithms instead of Euclidean ...

We look at complex numbers which are values of classical closed form expressions, and consider the relationship between the length of an expression and the number of decimal places needed to distinguish its absolute value from zero (if it is non-zero).

We accelerate the extended Euclidean algorithm for integers, the rational number reconstruction, and consequently, the stage of the recovery of the solution of a nonsingular integer system of linear equations via Hensel's lifting. The acceleration is by ...

Polynomial factorization plays a significant role in computational mathematics and its application to engineering, since it forms a fundamental part of algorithms for higher algebraic computations. Multivariate polynomial factorization over finite ...

Let F(x, u) be a given bivariate polynomial and ε be a small positive number. We consider the approximate factorization of F: find polynomials G, H and Δ_F such that F = GH + Δ_F and ‖Δ_F‖ / ‖F‖= ε, where ‖P‖ denotes 2-norm of polynomial P. At first, we ...

We present a probabilistic seminumerical algorithm that computes the differential Hilbert function associated to a differential rational mapping. This algorithm explicitly determines the set of variables and derivatives which can be arbitrarily fixed in ...

We present an algorithm for factoring a zero-dimensional left ideal in the ring Q(x, y) [∂_x, ∂_y], i.e. factoring a linear homogeneous partial differential system whose coefficients belong to Q(x, y), and whose solution space is finite-dimensional over ...

We propose a strategy for simplification of definite sums of rational functions which, for a given input rational function F(n, k), constructs two rational functions G(n) and T(n, k) such that∑_k=0ⁿF(n, k) = G(n) + ∑_k=0ⁿT(n, k),where the degree of the ...

In this paper we conduct a careful study of the equivalence classes of ternary cubics under general complex linear changes of variables. Our new results are based on the method of moving frames and involve triangular decompositions of algebraic ...

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 Sylvester and ...

In the first half of 2000, two new algorithms were discovered for the efficient computation of the determinant of a (dense) matrix with integer entries. Suppose that the dimension of the matrix is n x n and that the maximum bit length of all entries is ...

We show that every perturbation A(λ, ε) of an n x n matrix polynomial A(λ) such that det A(λ) = λ^m with m ≤ n can be reduced by equivalence transforms to a perturbed matrix polynomial whose leading matrix has maximal Smith form. This yields a reduced ...

The knowledge of sparsity information plays an important role in efficient determination of sparse Jacobian matrices. In a recent work, we have proposed sparsity-exploiting substitution techniques to determine Jacobian matrices. In this paper, we take a ...

In this paper, we present a new zero-test for expressions which are constructed from formal power solutions to algebraic differential equations using the ring operations and differentiation. We also provide a survey of all existing methods that we know ...