[Paper Review] Univariate Contraction and Multivariate Desingularization of Ore Ideals

Yi Zhang|arXiv (Cornell University)|Oct 20, 2017

Polynomial and algebraic computation28 references3 citations

TL;DR

This paper introduces a novel algorithm for computing the univariate contraction of Ore ideals in $ R[x][\partial] $, where $ R $ is a principal ideal domain, using desingularized operators. It proposes the concept of completely desingularized operators and provides a method to detect and remove apparent singularities in multivariate D-finite systems via Gröbner basis techniques, enabling certification of integer sequences and verification of special cases of Krattenthaler's conjecture.

ABSTRACT

Ore operators with polynomial coefficients form a common algebraic abstraction for representing D-finite functions. They form the Ore ring $K(x)[D_x]$, where $K$ is the constant field. Suppose $K$ is the quotient field of some principal ideal domain $R$. The ring $R[x][D_x]$ consists of elements in $K(x)[D_x]$ without "denominator". Given $L \in K(x)[D_x]$, it generates a left ideal $I$ in $K(x)[D_x]$. We call $I \cap R[x][D_x]$ the univariate contraction of $I$. When $L$ is a linear ordinary differential or difference operator, we design a contraction algorithm for $L$ by using desingularized operators as proposed by Chen, Jaroschek, Kauers and Singer. When $L$ is an ordinary differential operator and $R = K$, our algorithm is more elementary than known algorithms. In other cases, our results are new. We propose the notion of completely desingularized operators, study their properties, and design an algorithm for computing them. Completely desingularized operators have interesting applications such as certifying integer sequences and checking special cases of a conjecture of Krattenthaler. A D-finite system is a finite set of linear homogeneous partial differential equations in several variables, whose solution space is of finite dimension. For such systems, we give the notion of a singularity in terms of the polynomials appearing in them. We show that a point is a singularity of the system unless it admits a basis of power series solutions in which the starting monomials are as small as possible with respect to some term order. Then a singularity is apparent if the system admits a full basis of power series solutions, the starting terms of which are not as small as possible. We prove that apparent singularities in the multivariate case can be removed like in the univariate case by adding suitable additional solutions to the original system.

Motivation & Objective

• To develop an algorithm for computing the univariate contraction of an Ore ideal $ I \cap R[x][\partial] $, where $ I $ is generated by a linear ordinary differential or difference operator in $ \mathbb{K}(x)[\partial] $.
• To define and compute completely desingularized operators with minimal degree and content, enabling improved algebraic manipulation of D-finite functions.
• To characterize singularities and apparent singularities in multivariate D-finite systems using formal power series solutions and term orderings.
• To provide an algorithmic method for detecting and removing apparent singularities in multivariate D-finite systems by adding suitable solutions.
• To lay the foundation for extending univariate contraction techniques to the multivariate Ore algebra $ R[\mathbf{x}][\boldsymbol{\partial}] $, with open problems on general upper bounds for contraction ideals.

Proposed method

• Use of desingularized operators as proposed by Chen, Jaroschek, Kauers, and Singer to enable denominator-free representation in $ R[x][\partial] $.
• Computation of Gröbner bases over the principal ideal domain $ R $ to determine contraction ideals and analyze initial exponent candidates.
• Definition of initial exponent candidates $ S $ based on the index of each generator in a Gröbner basis, using term orderings on monomials.
• Construction of a new Gröbner basis $ M $ by intersecting the original ideal with ideals generated by $ \{x_i\partial_i - s_i\} $ for $ (s_1, \dots, s_n) \notin S $.
• Use of the head characteristic (HC) of the resulting Gröbner basis $ M $ to determine whether a singularity is apparent: if $ \operatorname{HC}(M) $ contains a non-zero constant, the point is ordinary.
• Application of Theorem 4.4.6 to classify singularities: a point is an apparent singularity if a full basis of power series solutions exists but with non-minimal starting monomials.

Experimental results

Research questions

• RQ1 How can the univariate contraction of an Ore ideal $ I \cap R[x][\partial] $ be computed efficiently for linear ordinary differential or difference operators?
• RQ2 What is the structure and algorithmic computation of completely desingularized operators, and how do they improve the representation of D-finite functions?
• RQ3 How can singularities in multivariate D-finite systems be characterized algebraically using formal power series solutions and term orderings?
• RQ4 In what way can apparent singularities in multivariate D-finite systems be algorithmically detected and removed?
• RQ5 What are the general upper bounds for the orders of generators of contraction ideals in the multivariate case, and how can they be computed?

Key findings

• The algorithm for univariate contraction is more elementary than known methods when $ R = \mathbb{K} $, the constant field, and applies to both differential and difference operators.
• Completely desingularized operators are introduced and shown to be effective for certifying integer sequences and verifying special cases of Krattenthaler’s conjecture.
• A point is a singularity of a D-finite system if no basis of power series solutions exists with starting monomials as small as possible under a given term order.
• An apparent singularity occurs when such a basis exists but with non-minimal starting monomials, and such singularities can be removed by adding appropriate solutions to the system.
• The origin is an apparent singularity of the system $ G $ in Example 4.4.14 because $ \operatorname{HC}(M) = \{-2 - x_1^2 - 2x_1x_2 - x_2^2\} $, indicating an ordinary point for $ M $, while $ G $ has non-minimal starting terms.
• The origin is not an apparent singularity of the system $ G $ in Example 4.4.13 because $ \operatorname{HC}(M) $ contains non-constant polynomials, implying the singularity is genuine.