A new perspective on single and multi-variate differential equations

We elaborate upon a new method of solving linear differential equations, of arbitrary order, which is applicable to a wide class of single and multi-variate equations. Our procedure separates the operator part of the equation under study in to a part containing a function of the Euler operator and constants, and another one retaining the rest. The solution of the equation is then obtained from the monomials (or the monomial symmetric functions, for the multi-variate case), which are the eigenfunctions of the Euler operator. Novel exponential forms of the solutions of the differential equations enable one to analyze the underlying symmetries of the equations and explore the algebraic structures of the solution spaces in a straightforward manner. The procedure allows one to derive various properties of the orthogonal polynomials and functions in a unified manner. After showing how the generating functions and Rodriguez formulae emerge naturally in this method, we briefly outline the generalization of the present approach to the multi-variate case.     

Coefficient Ideal of Ideals Generated by Monomials

In a commutative Noetherian ring R, the coefficient ideal of I relative to J is the largest ideal 𝔟 for which I𝔟 =J𝔟 when I is integral over J. In this article, we will give a simple algorithm to compute 𝔞(I, J) when I, J are ideals in a polynomial ring R = k[X ₁,…, X _d ] generated by monomials and J is a parameter ideal. We use the concept of socle sequence. Also we will show that the reduction number r _J (I) is also computed by our algorithm.     

The Jet Scheme of a Monomial Scheme

We explicitly compute the equations and components of the jet schemes of a monomial subscheme of affine space from an algebraic perspective.