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The level of pairs of polynomials. (English) Zbl 1444.13008
Let $$k$$ be any perfect field. Set $$R:=k[x_1,\ldots ,x_d]$$ and let $$\mathcal{D}_R$$ be the ring of $$k$$-linear differential operators on $$R$$. For a non-zero $$f\in R$$, let $$R_f$$ be the localization of $$R$$ at $$f$$. The natural action of $$\mathcal{D}_R$$ on $$R$$ extends uniquely to $$R_f$$ and it is known that there exists $$m\geq 1$$ such that $$R_f=\mathcal{D}_R\frac{1}{f^m}$$. Given a polynomial $$f$$ with coefficients in a field of prime characteristic $$p$$, it is known that there exists a differential operator that raises $$1/f$$ to its $$p$$th power. The authors first discuss a relation between the “level” of this differential operator and the notion of “stratification” in the case of hyperelliptic curves. They extend the notion of level to that of a pair of polynomials, they prove some basic properties and compute this level in certain special cases. They present examples of polynomials $$g$$ and $$f$$ such that there is no differential operator raising $$g/f$$ to its $$p$$th power.
##### MSC:
 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure 13N10 Commutative rings of differential operators and their modules 14B05 Singularities in algebraic geometry 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
##### Software:
Magma; TestIdeals
Full Text:
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