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This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present…

We describe our ongoing project of formalization of algebraic methods for geometry theorem proving (Wu's method and the Groebner bases method), their implementation and integration in educational tools. The project includes formal…

We consider the problem of computing critical points of the restriction of a polynomial map to an algebraic variety. This is of first importance since the global minimum of such a map is reached at a critical point. Thus, these points…

We analyze the differential equations produced by the method of creative telescoping applied to a hyperexponential term in two variables. We show that equations of low order have high degree, and that higher order equations have lower…

Sparse (or toric) elimination exploits the structure of polynomials by measuring their complexity in terms of Newton polytopes instead of total degree. The sparse, or Newton, resultant generalizes the classical homogeneous resultant and its…

We show that the problem of constructing telescopers for functions of m variables is equivalent to the problem of constructing telescopers for algebraic functions of m -1 variables and present a new algorithm to construct telescopers for…

Creative telescoping applied to a bivariate proper hypergeometric term produces linear recurrence operators with polynomial coefficients, called telescopers. We provide bounds for the degrees of the polynomials appearing in these operators.…

Elementary tutorial on implementation aspects of Gr\"obner bases computation.

We present an algorithm for tests generation tools based on symbolic execution. The algorithm is supposed to help in situations, when a tool is repeatedly failing to cover some code by tests. The algorithm then provides the tool a necessary…

The classical division algorithm for polynomials requires $O(n^2)$ operations for inputs of size $n$. Using reversal technique and Newton iteration, it can be improved to $O({M}(n))$, where ${M}$ is a multiplication time. But the method…

Kaltofen has proposed a new approach in [Kaltofen 1992] for computing matrix determinants. The algorithm is based on a baby steps/giant steps construction of Krylov subspaces, and computes the determinant as the constant term of a…

Let $f_1,...,f_s \in \mathbb{K}[x_1,...,x_m]$ be a system of polynomials generating a zero-dimensional ideal $\I$, where $\mathbb{K}$ is an arbitrary algebraically closed field. We study the computation of "matrices of traces" for the…

In 1977, Adleman, Manders and Miller had briefly described how to extend their square root extraction method to the general $r$th root extraction over finite fields, but not shown enough details. Actually, there is a dramatic difference…

This note presents absolute bounds on the size of the coefficients of the characteristic and minimal polynomials depending on the size of the coefficients of the associated matrix. Moreover, we present algorithms to compute more precise…

The extended L\"uroth's Theorem says that if the transcendence degree of $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK$ is 1 then there exists $f \in \KK(\underline{X})$ such that $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)$ is equal to $\KK(f)$. In…

Root extraction is a classical problem in computers algebra. It plays an essential role in cryptosystems based on elliptic curves. In 2006, Barreto and Voloch proposed an algorithm to compute $r$th roots in ${F}_{q^m} $ for certain choices…

The foundational theory of differentiation was developed as part of the original release of ACL2(r). In work reported at the last ACL2 Workshop, we presented theorems justifying the usual differentiation rules, including the chain rule and…

We generalize the structural theorem of Lazard in 1985, from 2 variables to 3 variables. We use the Gianni-Kalkbrener result to do this, which implies some restrictions inside which lies the case of a radical ideal.

We develop symbolic methods of asymptotic approximations for solutions of linear ordinary differential equations and use to them stabilize numerical calculations. Our method follows classical analysis for first-order systems and…

Let $a_n$ denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an $n \times n \times n$ three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step.…