This paper is devoted to the complexity analysis of certain uniformity properties owned by all known symbolic methods of parametric polynomial equation solving (geometric elimination). It is shown that any parametric elimination procedure which is parsimonious with respect to branchings and divisions must necessarily have a non-polynomial sequential time complexity, even if highly ecient data structures (as e.g. the arithmetic circuit encoding of polynomials) are used.

In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)-tuple P = ( f, g1, g2,..., gw) where f and the gi are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the gi. For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert’s Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases.

Over the past twenty years rapid advances in computational algebraic geometry have generated increasing amounts of interest in quantifying the “complexity ” of ideals and modules. For a finitely generated module M over a polynomial ring S = k[x0,..., xn] with k an arbitrary field, we say that M is r-regular (in the

Many applications of constraint programming languages concern geometric domains. We propose incorporating strong algorithmic techniques from the study of geometric and algebraic algorithms into the implementation of constraint programming languages. Interesting new computational problems in computational geometry and computer algebra arises from such considerations. We look at what is known and what needs to be addressed.

We develop a probabilistic test for the vanishing of radical expressions, that is, expressions involving the four rational operations (+; ; ; ) and square root extraction. This extends the well-known Schwartz's probabilistic test for the vanishing of polynomials. The probabilistic test forms the basis of a new theorem prover for conjectures about ruler & compass constructions. Our implementation uses the Core Library which can perform exact comparison for radical expressions.

A polynomial ideal membership problem is a (w+1)-tuple P = (f; g 1 ; g 2 ; : : : ; g w ) where f and the g i are multivariate polynomials over some ring, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, it is known that this problem is exponential space complete.