In this paper,we consider symbolic(hybrid trigonometric)parametrizations defined as tuples of real rational expressions involving circular and hyperbolic trigonometric functions as well as monomials,with the restricti...In this paper,we consider symbolic(hybrid trigonometric)parametrizations defined as tuples of real rational expressions involving circular and hyperbolic trigonometric functions as well as monomials,with the restriction that variables in each block of functions are different.We prove that the varieties parametrizable in this way are exactly the class of real unirational varieties of any dimension.In addition,we provide symbolic algorithms to implicitize and to convert a hybrid trigonometric parametrization into a unirational one,and vice versa.We illustrate by some examples the applicability of having these different types of parametrizations,namely,hybrid trigonometric and unirational.展开更多
基金the CRUE-CSIC agreement with Springer Nature.the project ID2019-105621GB-I00 of Ministerio de Ciencia e Innovación.+1 种基金the GrantPID2020-113192GB-100(Math-ematical Visualization:Foundations,Algorithms and Applications)from the Spanish MICINN.A.Lastra is member of the Research Group ASYNACS(Ref.CT-CE2019/683).
文摘In this paper,we consider symbolic(hybrid trigonometric)parametrizations defined as tuples of real rational expressions involving circular and hyperbolic trigonometric functions as well as monomials,with the restriction that variables in each block of functions are different.We prove that the varieties parametrizable in this way are exactly the class of real unirational varieties of any dimension.In addition,we provide symbolic algorithms to implicitize and to convert a hybrid trigonometric parametrization into a unirational one,and vice versa.We illustrate by some examples the applicability of having these different types of parametrizations,namely,hybrid trigonometric and unirational.
