Singular Places and Indicial Polynomials#

All functions treated in this section concern differential operator rings defined over a function field of transcendence degree one. They all require that the derivation of such an operator ring \(F[D]\) over \(F\) is defined with respect to a differential.

Any solution of a differential operator \(L\in F[D]\) is also a solution of the operator \(c\cdot L\), for any non–zero \(c\in F\). For considering solutions we may therefore consider \(L\) to be monic of the form

\[L = D^n+a_{n-1}D^{n-1}+\cdots+a_0,\]

with coefficients \(a_i\in F\) for \(i=1,2,\ldots n-1\). Each of the coefficients is a rational function in \(F\). They play an important role in the rational solutions of \(L(y)=0\) in \(F\) that come to expression in so-called regular and singular places.

Singular Places#

Given a place \((q)\) in the set of places of \(F\), a putative rational solution \(y\in F\) of \(L(y)=0\) has a \(q\)-adic expansion

\[y = y_{\alpha}\, q^{ \alpha}+ y_{\alpha+1}\, q^{\alpha+1}+\ldots\]

It has a pole at the place \((q)\) if the valuation \(\alpha\) is negative. After substituting this solution in \(L(y)\) it becomes clear that there is only a finite number of places which can occur as poles of an arbitrary solution of \(Ly=0\). Such a place is either a pole of one of the coefficients \(a_i\) or a zero or a pole of the differential \(\omega\) of \(F[D]\). There exists a classification for the poles of solutions of \(L(y)=0\).

Given a place \((q)\) and local parameter \(t\) at \((q)\), the differential operator can be rewritten as a differential operator \(\tilde{L}\in \tilde{F}[\tilde{D}]\), with \(\tilde{F}\cong F\), the valuation of whose differential at \((q)\) is \(0\). The place \((q)\) is defined to be a singular place of \(L\), if one of the coefficients of \(\tilde{L}\) has negative valuation at \((q)\). Places that are not singular are called regular.

There are two kinds of singular places of a differential operator; the regular singular places and the irregular singular places. With the notation as above, a singular place \((q)\) of \(L\) is regular singular if the valuation of the coefficient of \((\tilde{D})^i\) in \(\tilde{L}\) is at most \(i-n\) for every \(i\in\{0,1,\ldots,n-1\}\). Otherwise it is irregular singular.

IsRegularPlace(L, p): RngDiffOpElt, PlcFunElt -> BoolElt#: Returns true iff the place \(p\) is a regular place of the differential operator \(L\). If \(p\) is not a regular place of \(L\) false is returned. This function only works for operators whose derivation is defined by a differential.

IsRegularSingularPlace(L, p): RngDiffOpElt, PlcFunElt -> BoolElt#: Returns true iff the place \(p\) is a regular singular place of the differential operator \(L\). If \(p\) is not a regular singular place of \(L\) false is returned. This function only works for operators whose derivation is defined by a differential.

IsIrregularSingularPlace(L, p): RngDiffOpElt, PlcFunElt -> BoolElt#: Returns true iff the place \(p\) is an irregular singular place of the differential operator \(L\). If \(p\) is not an irregular singular place of \(L\) false is returned. This function only works for operators whose derivation is defined by a differential.

SetsOfSingularPlaces(L): RngDiffOpElt -> SetEnum, SetEnum#: Two sets are returned. The first set contains precisely all regular singular places of the differential operator \(L\). The second set consists of all irregular singular places of \(L\). This function only works for operators whose derivation is defined by a differential.

IsFuchsianOperator(L): RngDiffOpElt -> BoolElt, SetEnum#: Returns true iff the differential operator \(L\) is Fuchsian (i.e. if all singular places of \(L\) are regular singular). If \(L\) is not Fuchsian, false is returned. Secondly, the set of all singular places of \(L\), is returned only if \(L\) is a Fuchsian differential operator. This function only works for operators whose derivation is defined by a differential.

IsRegularSingularOperator(L): RngDiffOpElt -> BoolElt, SetEnum#: Returns true if and only if the differential operator \(L\) is regular singular. The operator may be defined over a differential Laurent series ring, \(F\) say. In this case being regular singular means that the operator must be regular singular at \(F.1\). Then also there is no second argument returned. In the case that the derivation is defined with respect to a differential, then the values from IsFuchsianOperator(L) are returned.

Example Singularities (ex-3dba78)#

> F<z> := RationalDifferentialField(Rationals());
> R<D> := DifferentialOperatorRing(F);
> H := (z^2-z)*D^2+(3*z-6)*D+1;
> IsRegularPlace(H, Zeros(z)[1]);
false
> IsRegularSingularPlace(H, Zeros(z)[1]);
true
> SetsOfSingularPlaces(H);
{ (1/z), (z - 1), (z) }
{}
> IsFuchsianOperator(H);
true { (z), (1/z), (z - 1) }
> IsFuchsianOperator(D^2-1/z^3);
false

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Example Regular Singular DLSR (ex-5e64d4)#

> S<t> := DifferentialLaurentSeriesRing(Rationals());
> R<D> := DifferentialOperatorRing(S);
> IsRegularSingularOperator(D^2 -t*D+2);
true
> IsRegularSingularOperator(D^2 +3);
true
> IsRegularSingularOperator(D^2 +3 +O(t));
true
> IsRegularSingularOperator(D^2 +3*t^(-1));
false

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Indicial Polynomials#

For the definition of a indicial polynomial at a place, we refer to Section \(4.1\) in [van der Put and Singer, 2003].

IndicialPolynomial(L, p): RngDiffOpElt, PlcFunElt -> RngElt#: Returns the monic indicial polynomial of the differential operator \(L\) at the place \(p\). This function only works for operators whose derivation is defined by a differential and whose base ring has one generator.

Example Indicial Pol (ex-eadf1f)#

> F<z> := RationalDifferentialField(Rationals());
> _<T> := PolynomialRing(Rationals());
> R<D> := DifferentialOperatorRing(F);
> H := (z^2-z)*D^2+(3*z-6)*D+1;
> IndicialPolynomial(H, Zeros(z)[1]);
T^2 + 5*T
> IndicialPolynomial(H, Zeros(z-1)[1]);
T^2 - 4*T
> IndicialPolynomial(H, Zeros(1/z)[1]);
T^2 - 2*T + 1
> Apply(H, (z-1)^4/z^5);
0

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