Related papers: Formal proofs in real algebraic geometry: from ord…
We construct a convenient basis for all real semisimple Lie algebras by means of an adapted Chevalley basis of the complexification. It determines rational and in fact half-integer structure constants which we express only in terms of the…
We give algorithms for the computation of the algebraic de Rham cohomology of open and closed algebraic sets inside projective space or other smooth complex toric varieties. The methods, which are based on Gr\"obner basis computations in…
In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
We introduce a formal framework for analyzing trades in financial markets. An exchange is where multiple buyers and sellers participate to trade. These days, all big exchanges use computer algorithms that implement double sided auctions to…
Whereas proof assistants based on Higher-Order Logic benefit from external solvers' automation, those based on Type Theory resist automation and thus require more expertise. Indeed, the latter use a more expressive logic which is further…
Reliably determining system trajectories is essential in many analysis and control design approaches. To this end, an initial value problem has to be usually solved via numerical algorithms which rely on a certain software realization.…
We describe new algorithms to compute Whitney stratifications of real algebraic varieties. Using either conormal or polar techniques, these algorithms stratify a complexification of a given real variety. We then show that the resulting…
When using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is likely not the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier…
We give a purely algebraic treatment of reduction theory for connections over the formal punctured disc. Our proofs apply to arbitrary connected linear algebraic groups over an algebraically closed field of characteristic 0. We also state…
We present a formally verified global optimization framework. Given a semialgebraic or transcendental function $f$ and a compact semialgebraic domain $K$, we use the nonlinear maxplus template approximation algorithm to provide a certified…
Computational content encoded into constructive type theory proofs can be used to make computing experiments over concrete data structures. In this paper, we explore this possibility when working in Coq with chain complexes of infinite type…
We report on the development of an optimized and verified decision procedure for orthologic equalities and inequalities. This decision procedure is quadratic-time and is used as a sound, efficient and predictable approximation to classical…
Computer Algebra systems are widely spread because of some of their remarkable features such as their ease of use and performance. Nonetheless, this focus on performance sometimes leads to unwanted consequences: algorithms and computations…
We study finite-dimensional groups definable in models of the theory of real closed fields with a generic derivation (also known as CODF). We prove that any such group definably embeds in a semialgebraic group. We extend the results to…
The canonical projections of the unit spheres are generalized to special generic maps and round fold maps, for example. They are generalizations from the viewpoint of singularity theory of differentiable maps and these maps restrict the…
Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…
Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an…
Initial Semantics aims at characterizing the syntax associated to a signature as the initial object of some category. We present an initial semantics result for typed higher-order syntax together with its formalization in the Coq proof…
In this monograph, we lay the foundations for a new theory that generalizes real algebraic geometry. Let $R|K$ be a field extension, where $R$ is a real closed field and $K$ is an ordered subfield of $R$. The main objective is to study…