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Register a new userDesingularization of First Order Linear Difference Systems with Rational Function Coefficients
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Added by
Maximilian Jaroschek
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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It is well known that for a first order system of linear difference equations with rational function coefficients, a solution that is holomorphic in some left half plane can be analytically continued to a meromorphic solution in the whole complex plane. The poles stem from the singularities of the rational function coefficients of the system. Just as for differential equations, not all of these singularities necessarily lead to poles in solutions, as they might be what is called removable. In our work, we show how to detect and remove these singularities and further study the connection between poles of solutions and removable singularities. We describe two algorithms to (partially) desingularize a given difference system and present a characterization of removable singularities in terms of shifts of the original system.
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Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An order-degree curve for a given Ore operator is a curve in the $(r,d)$-plane such that for all points $(r,d)$ above this curve, there exists a left multiple of order $r$ and degree $d$ of the given operator. We give a new proof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization implies order-degree curves which are extremely accurate in examples.
The detailed construction of the general solution of a second order non-homogenous linear operatordifference equation is presented. The wide applicability of such an equation as well as the usefulness of its resolutive formula is shown by studying some applications belonging to different mathematical contexts.
The exact solution of a Cauchy problem related to a linear second-order difference equation with constant noncommutative coefficients is reported.
Let L be a linear difference operator with polynomial coefficients. We consider singularities of L that correspond to roots of the trailing (resp. leading) coefficient of L. We prove that one can effectively construct a left multiple with polynomial coefficients L of L such that every singularity of L is a singularity of L that is not apparent. As a consequence, if all singularities of L are apparent, then L has a left multiple whose trailing and leading coefficients equal 1.
This is a self-contained purely algebraic treatment of desingularization of fields of fractions $mathbf{L}:=Q(mathbf{A})$ of $d$-dimensional domains of the form [mathbf{A}:=bar{mathbf{F}}[underline{x}]/langle b(underline{x})rangle] with a purely algebraic objective of uniquely describing $d$-dimensional valuations in terms of $d$ explicit (independent) local parameters and $1$ (dependent) local unit, for arbitrary dimension $d$ and arbitrary characteristic $p$. The desingularization will be given as a rooted tree with nodes labelled by domains $mathbf{A}_k$ (all with field of fractions $Q(mathbf{A}_k)=mathbf{L}$), sets $EQ_k$ and $INEQ_k$ of equality constraints and inequality constraints, and birational change-of-variables maps on $mathbf{L}$. The approach is based on d-dimensional discrete valuations and local monomial orderings to emphasize formal Laurent series expansions in $d$ independent variables. It is non-standard in its notation and perspective.
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