# Computational Algebra and Geometric Modeling (16w5115)

Arriving in Oaxaca, Mexico Sunday, August 7 and departing Friday August 12, 2016

## Organizers

Laurent Buse (Institute for Research in Computer Science and Automation)

Ron Goldman (Rice University)

Hal Schenck (University of Illinois Urbana Champaign)

## Objectives

The interactions between geometric modeling and algebraic geometry are driven by natural developments in each field. In algebraic geometry it is the study of computational methods while in geometric modeling, it is the drive to master and use more theoretical tools in the study of their basic objects. Let us give a short list (non exhaustive) of some particular topics of strong collaboration.

The original interactions between the subjects were mostly algebraic. In modeling, the basic objects are parametric curves and surfaces. Determining intersection between these objects is a fundamental problem in geometric modeling. Intersections require computing an implicit representation of the parametric object, which is a very difficult and subtle algebraic problem. In the 1980's Sederberg realized that the classical Dixon resultant solves this problem for general parametric surfaces. For the less general surfaces that often arise, significantly more subtle methods are needed, and the method of moving lines, (and planes) was developed, which turns out to be an innovative way to compute syzygies in computational algebraic geometry. There is a very well-developed connection and flow of ideas between the subjects related to resultants and syzygies, whose impact reaches beyond the original areas of interaction. This conference will be an occasion to shed light on recent developments in the use of multi-graded parameterizations which lead to finer implicit representations of curves and surfaces through the study of algebraic properties of some associated symmetric algebras.

Toric geometry is also an important area of collaboration between algebraic geometry and geometric modeling. Krasauskas realized that the basic objects in geometric modeling, Bezier curves and surfaces, in fact come naturally from an important class of algebraic varieties called toric varieties. The impact of this insight for geometric modeling has been major : it has led to a deeper understanding of these basic objects in modeling, and a study of the properties of Krasauskas' toric patches. These patches come from the positive part of a toric variety, which is an object common to several additional fields such as approximation theory. Approximation theory is actually related to modeling via barycentric coordinates and splines, and it also leads to interesting interactions with algebraic geometry. For example, questions such as accuracy, precision and rate of convergence are considered in geometric modeling as well as in approximation theory. The matter of (linear) precision in toric algebraic geometry turned out to be a related tool -- more collaboration is definitely needed in this area with potential benefits to the approximation theory community. A more recent interaction that we hope to develop further during this conference is the interplay between solving real polynomial systems and geometric modeling. Indeed, geometric modeling often needs to find real roots of equations, or to solve inequalities. It also needs to determine shape from constraints. Current methods often reduce these questions to the solutions of many thousands of systems of polynomial equations. Fortunately, algebraic geometry has been developing theoretical and practical tools for understanding and computing the real solutions to systems of equations and understanding systems of inequalities.

All the above-mentioned topics converge in some traditional fundamental problems which are still under investigation such as the detection and analysis of the singularities of curves and surfaces and that will definitely benefit from the interaction between algebraic geometry and geometric modeling. Besides all these specific topical scientific objectives, the main objective of this international conference is to continue to engage in the hard work of bringing all these different communities together and we believe that a full week of collaboration at CIRM would help to sustain this trend.

The original interactions between the subjects were mostly algebraic. In modeling, the basic objects are parametric curves and surfaces. Determining intersection between these objects is a fundamental problem in geometric modeling. Intersections require computing an implicit representation of the parametric object, which is a very difficult and subtle algebraic problem. In the 1980's Sederberg realized that the classical Dixon resultant solves this problem for general parametric surfaces. For the less general surfaces that often arise, significantly more subtle methods are needed, and the method of moving lines, (and planes) was developed, which turns out to be an innovative way to compute syzygies in computational algebraic geometry. There is a very well-developed connection and flow of ideas between the subjects related to resultants and syzygies, whose impact reaches beyond the original areas of interaction. This conference will be an occasion to shed light on recent developments in the use of multi-graded parameterizations which lead to finer implicit representations of curves and surfaces through the study of algebraic properties of some associated symmetric algebras.

Toric geometry is also an important area of collaboration between algebraic geometry and geometric modeling. Krasauskas realized that the basic objects in geometric modeling, Bezier curves and surfaces, in fact come naturally from an important class of algebraic varieties called toric varieties. The impact of this insight for geometric modeling has been major : it has led to a deeper understanding of these basic objects in modeling, and a study of the properties of Krasauskas' toric patches. These patches come from the positive part of a toric variety, which is an object common to several additional fields such as approximation theory. Approximation theory is actually related to modeling via barycentric coordinates and splines, and it also leads to interesting interactions with algebraic geometry. For example, questions such as accuracy, precision and rate of convergence are considered in geometric modeling as well as in approximation theory. The matter of (linear) precision in toric algebraic geometry turned out to be a related tool -- more collaboration is definitely needed in this area with potential benefits to the approximation theory community. A more recent interaction that we hope to develop further during this conference is the interplay between solving real polynomial systems and geometric modeling. Indeed, geometric modeling often needs to find real roots of equations, or to solve inequalities. It also needs to determine shape from constraints. Current methods often reduce these questions to the solutions of many thousands of systems of polynomial equations. Fortunately, algebraic geometry has been developing theoretical and practical tools for understanding and computing the real solutions to systems of equations and understanding systems of inequalities.

All the above-mentioned topics converge in some traditional fundamental problems which are still under investigation such as the detection and analysis of the singularities of curves and surfaces and that will definitely benefit from the interaction between algebraic geometry and geometric modeling. Besides all these specific topical scientific objectives, the main objective of this international conference is to continue to engage in the hard work of bringing all these different communities together and we believe that a full week of collaboration at CIRM would help to sustain this trend.