Rational plane curves parameterizable by conics

We introduce the class of rational plane curves parameterizable by conics as an extension of the family of curves parameterizable by lines (also known as monoid curves). We show that they are the image of monoid curves via suitable quadratic transformations in projective plane. We also describe all the possible proper parameterizations of them, and a set of minimal generators of the Rees Algebra associated to these parameterizations, extending well-known results for curves parameterizable by lines.Comment: 28 pages, 1 figure. Revised version. Accepted for publication in Journal of Algebr

Computing the μ-bases of algebraic monoid curves and surfaces

The μ-basis is a developing algebraic tool to study the expressions of rational curves and surfaces. It can play a bridge role between the parametric forms and implicit forms and show some advantages in implicitization, inversion formulas and singularity computation. However, it is difficult and there are few works to compute the μ-basis from an implicit form. In this paper, we derive the explicit forms of μ-basis for implicit monoid curves and surfaces, including the conics and quadrics which are particular cases of these entities. Additionally, we also provide the explicit form of μ-basis for monoid curves and surfaces defined by any rational parametrization (not necessarily in standard proper form). Our technique is simply based on the linear coordinate transformation and standard forms of these curves and surfaces. As a practical application in numerical situation, if an exact multiple point can not be computed, we can consider the problem of computing “approximate μ-basis” as well as the error estimation.Agencia Estatal de Investigació

Ideals de corbes mòbils i la seva interacció amb el disseny assistit per ordinador

Presentem un cas d'interacció fructífera entre l'àlgebra commutativa i el disseny assistit per ordinador. Els problemes en aquesta àrea aplicada i cada vegada més important de la informàtica s'han traslladat a l'estudi d'estructures algebraiques abstractes, i han enriquit la matemàtica amb múltiples resultats teòrics i problemes oberts que expliquem en aquest text

CANONICAL BASES FOR CLUSTER ALGEBRAS

In an earlier work (Publ. Inst. Hautes Études Sci., 122 (2015), 65–168) the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi–Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalizations of holomorphic discs). Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock–Goncharov dual basis conjecture (Publ. Inst. Hautes Études Sci., 103 (2006), 1–211). In particular, under suitable hypotheses, for each Y Y the partial compactification of an affine cluster variety U U given by allowing some frozen variables to vanish, we obtain canonical bases for H 0 ( Y , O Y ) H^0(Y,\mathcal {O}_Y) extending to a basis of H 0 ( U , O U ) H^0(U,\mathcal {O}_U) . Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell U U in the basic affine space Y , Y, we obtain a canonical basis of each irreducible representation of SL r \operatorname {SL}_r , parameterized by a set which each choice of seed identifies with the integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation-theoretic considerations. Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.The first author was partially supported by NSF grant DMS-1262531 and a Royal Society Wolfson Research Merit Award, the second by NSF grants DMS-1201439 and DMS-1601065, and the third by NSF grant DMS-0854747