Analogue of Newton–puiseux Series for Non-holonomic D-modules and Factoring
نویسندگان
چکیده
We introduce a concept of a fractional derivatives series and prove that any linear partial differential equation in two independent variables has a fractional derivatives series solution with coefficients from a differentially closed field of zero characteristic. The obtained results are extended from a single equation to D-modules having infinitedimensional space of solutions (i.e., non-holonomic D-modules). As applications we design algorithms for treating first-order factors of a linear partial differential operator, in particular for finding all (right or left) first-order factors. 2000 Math. Subj. Class. 35C10, 35D05, 68W30.
منابع مشابه
On the complexity of solving ordinary differential equations in terms of Puiseux series
We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials associated to ordinary linear differential operators. In this paper, we get the same bound for arbitrary differential polynomials. The algori...
متن کاملDynamic Newton–Puiseux Theorem
A constructive version of Newton–Puiseux theorem for computing the Puiseux expansions of algebraic curves is presented. The proof is based on a classical proof by Abhyankar. Algebraic numbers are evaluated dynamically; hence the base field need not be algebraically closed and a factorization algorithm of polynomials over the base field is not needed. The extensions obtained are a type of regula...
متن کاملPuiseux Series Solutions of Ordinary Polynomial Differential Equations : Complexity Study
We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given in 1990 by Grigoriev for Riccatti differential polynomials associated to ordinary linear differential operators. In this paper, we get the same bound for arbitrary differential polynomials. The alg...
متن کاملDynamic Newton–Puiseux Theorem
A constructive version of Newton–Puiseux theorem for computing the Puiseux expansions of algebraic curves is presented. The proof is based on a classical proof by Abhyankar. Algebraic numbers are evaluated dynamically; hence the base field need not be algebraically closed and a factorization algorithm of polynomials over the base field is not needed. The extensions obtained are a type of regula...
متن کاملPuiseux Power Series Solutions for Systems of Equations
We give an algorithm to compute term by term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton’s method for plane algebraic curves replacing the Newton polygon by the tropical variety of the ideal generated by the system.
متن کامل