Marc Moreno Maza

On the Theories of Triangular Sets

Different notions of triangular sets are presented. The relationship between these notions are studied. The main result is that four different existing notions of good triangular sets are equivalent.

Lifting techniques for triangular decompositions

We present lifting techniques for triangular decompositions of zero-dimensional varieties, that extend the range of the previous methods. We discuss complexity aspects, and report on a preliminary implementation. Our theoretical results are comforted by these experiments.

Computing cylindrical algebraic decomposition via triangular decomposition

Cylindrical algebraic decomposition is one of the most important tools for computing with semi-algebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set <i>F</i> ⊂ [<i>y₁</i>,...,<i>y_n</i>] we apply comprehensive triangular decomposition in order to obtain an <i>F</i>-invariant cylindrical decomposition of the <i>n</i>-dimensional complex space, from which we extract an <i>F</i>-invariant cylindrical algebraic decomposition of the <i>n</i>-dimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.

Polynomial Gcd Computations over Towers of Algebraic Extensions

Some methods for polynomial system solving require efficient techniques for computing univariate polynomial gcd over algebraic extensions of a field. Currently used techniques compute generic univariate polynomial gcd before specializing the result using algebraic relations in the ring of coefficients. This strategy generates very big intermediate data and fails for many problems. We present here a new approach which takes permanently into account those algebraic relations. It is based on a property of subresultant remainder sequences and leads to a great increase of the speed of computation and thus the size of accessible problems.

Triangular Sets for Solving Polynomial Systems

Four methods for solving polynomial systems by means of triangular sets are presented and implemented in a unified way. These methods are those of Wu (1987), Lazard (1991), Kalkbrener (1991) and Wang (1993b). They are compared on various examples with the emphasis on efficiency, conciseness and legibility of the output.

Algorithms for computing triangular decomposition of polynomial systems

We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions. We start with an overview of the key ideas which have led to either better implementation techniques or a better understanding of the underlying theory. We then present new techniques that we regard as essential to the recent success and for future research directions in the development of triangular decomposition methods.

Fast arithmetic for triangular sets: from theory to practice

We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, we reach quasi-linear complexity. The main outcome we have in mind is the acceleration of higher-level algorithms, by interfacing our low-level implementation with languages such as AXIOM or Maple We show the potential for huge speed-ups, by comparing two AXIOM implementations of van Hoeij and Monagans modular GCD algorithm.

Computations modulo regular chains

The computation of triangular decompositions involve two fundamental operations: polynomial GCDs modulo regular chains and regularity test modulo saturated ideals. We propose new algorithms for these core operations based on modular methods and fast polynomial arithmetic. We rely on new results connecting polynomial subresultants and GCDs modulo regular chains. We report on extensive experimentation, comparing our code to pre-existing Maple implementations, as well as more optimized Magma functions. In most cases, our new code outperforms the other packages by several orders of magnitude.

The modpn library: Bringing fast polynomial arithmetic into Maple

We investigate the integration of C implementation of fast arithmetic operations into Maple, focusing on triangular decomposition algorithms. We show substantial improvements over existing Maple implementations; our code also outperforms Magma on many examples. Profiling data show that data conversion can become a bottleneck for some algorithms, leaving room for further improvements.

Computation of canonical forms for ternary cubics

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 varieties. We provide a computationally efficient algorithm that matches an arbitrary ternary cubic with its canonical form and explicitly computes a corresponding linear change of coordinates. We also describe a classification of the symmetry groups of ternary cubics.