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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
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