Unidad docente Matemáticas http://hdl.handle.net/10017/281 MATEMATIC 2022-09-26T13:44:53Z On the base point locus of surface parametrizations: formulas and consequences http://hdl.handle.net/10017/51737 On the base point locus of surface parametrizations: formulas and consequences Cox, David A.; Pérez Díaz, Sonia; Sendra Pons, Juan Rafael This paper shows that the multiplicity of the base point locus of a projective rational surface parametrization can be expressed as the degree of the content of a univariate resultant. As a consequence, we get a new proof of the degree formula relating the degree of the surface, the degree of the parametrization, the base point multiplicity and the degree of the rational map induced by the parametrization. In addition, we extend both formulas to the case of dominant rational maps of the projective plane and describe how the base point loci of a parametrization and its reparametrizations are related. As an application of these results, we explore how the degree of a surface reparametrization is affected by the presence of base points. 2022-05-10T00:00:00Z Algebraic and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one in several variables http://hdl.handle.net/10017/51628 Algebraic and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one in several variables Cano, José; Falkensteiner, Sebastian; Robertz, Daniel; Sendra Pons, Juan Rafael In this paper we study systems of autonomous algebraic ODEs in several differential indeterminates. We develop a notion of algebraic dimension of such systems by considering them as algebraic systems. Afterwards we apply differential elimination and analyze the behavior of the dimension in the resulting Thomas decomposition. For such systems of algebraic dimension one, we show that all formal Puiseux series solutions can be approximated up to an arbitrary order by convergent solutions. We show that the existence of Puiseux series and algebraic solutions can be decided algorithmically. Moreover, we present a symbolic algorithm to compute all algebraic solutions. The output can either be represented by triangular systems or by their minimal polynomials. 2022-04-22T00:00:00Z Summability of formal solutions for a family of generalized moment integro-differential equations http://hdl.handle.net/10017/50614 Summability of formal solutions for a family of generalized moment integro-differential equations Lastra Sedano, Alberto; Michalik, Slawomir; Suwinska, Maria Generalized summability results are obtained regarding formal solutions of certain families of linear moment integro-differential equations with time variable coefficients. The main result leans on the knowledge of the behavior of the moment derivatives of the elements involved in the problem. A refinement of the main result is also provided giving rise to more accurate results which remain valid in wide families of problems of high interest in practice, such as fractional integro-differential equations. 2021-10-28T00:00:00Z A survey of the representations of rational ruled surfaces http://hdl.handle.net/10017/50574 A survey of the representations of rational ruled surfaces Yuan, Chun-Ming; Pérez Díaz, Sonia; Shen, Li-Yong The rational ruled surface is a typical modeling surface in computer aided geometric design. A rational ruled surface may have different representations with respective advantages and disadvantages. In this paper, the authors revisit the representations of ruled surfaces including the parametric form, algebraic form, homogenous form and Pl&#168;ucker form. Moreover, the transformations between these representations are proposed such as parametrization for an algebraic form, implicitization for a parametric form, proper reparametrization of an improper one and standardized reparametrization for a general parametrization. Based on these transformation algorithms, one can give a complete interchange graph for the different representations of a rational ruled surface. For rational surfaces given in algebraic form or parametric form not in the standard form of ruled surfaces, the characterization methods are recalled to identify the ruled surfaces from them. 2021-01-12T00:00:00Z