Related papers: Computing totally real hyperplane sections and lin…
This paper aims at settling the issue of the validity of the de Jonqui\`eres formulas. We consider the space of divisors with prescribed multiplicity, or de Jonqui\`eres divisors, contained in a linear series on a smooth projective curve.…
We provide criteria for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. These criteria reduce the projection problem to a certain…
Consider a real algebraic curve with set of real points $R\neq\emptyset$ and complexification $P\supset R$. Let $f$ be an algebraic function on $P$ with devisor of critical points $D\subset P$. We prove that $f$ is real after a…
Seeking tighter relaxations of combinatorial optimization problems, semidefinite programming is a generalization of linear programming that offers better bounds and is still polynomially solvable. Yet, in practice, a semidefinite program is…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
This paper first gives a brief overview over some interesting descriptions of conic sections, showing formulations in the three geometric algebras of Euclidean spaces, projective spaces, and the conformal model of Euclidean space. Second…
We study the number of hamiltonian circuits, containing a fixed basis, and the number of hyperplanes, which do not contain a fixed basis in perfect matroid designs. Projective and affine finite geometries are considered as examples of such…
The theory of Q-Cartier divisors on the space of n-pointed, genus 0, stable maps to projective space is considered. Generators and Picard numbers are computed. A recursive algorithm computing all top intersection products of Q-Divisors is…
A rational function on a real algebraic curve $C$ is called separating if it takes real values only at real points. Such a function defines a covering $\Bbb R C\to\Bbb{RP}^1$. Let $A_1,\dots,A_n$ be connected components of $C$. In a recent…
Let $C$ be a complex irreducible plane curve that is not the vanishing locus of a modular polynomial. We show that $C$ contains finitely many real algebraic curves whose projection on each coordinate axis is a union of special geodesics.
Let n be a positive integer, and let R be a finitely presented (but not necessarily finite dimensional) associative algebra over a computable field. We examine algorithmic tests for deciding (1) if every n-dimensional representation of R is…
We give practical algorithms for computing the divisor class group and the gonality of a curve over a finite field, achieving several orders of magnitude speedup over existing methods for sufficiently large genus or residue field. The…
We study integral points on affine surfaces by means of a new method, relying on the Subspace Theorem. Under suitable assumptions on the divisor at infinity, we prove that the integral points are contained in a curve. As a corollary, we…
Cutting a polytope is a very natural way to produce new classes of interesting polytopes. Moreover, it has been very enlightening to explore which algebraic and combinatorial properties of the orignial polytope are hereditary to its…
In this largely-expository note, we describe a class of divisors on elliptic curves that index the inflection points of linear series arising (as subspaces of holomorphic sections) from line bundles on $\mathbb{P}^1$ via pullback along the…
Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we…
Two plane analytic branches are topologically equivalent if and only if they have the same multiplicity sequence. We show that having same semigroup is equivalent to having same multiplicity sequence, we calculate the semigroup from a…
We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. A straightforward approach to this problem consists of…
We present an algorithm for detecting basepoints of linear series of curves in the plane. Moreover, we give an algorithm for constructing a linear series of curves in the plane for given basepoints. The underlying method of these algorithms…