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Register a new userOn symmetric partial differential operators
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Authors
Daniel Barlet
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Let s 1 ,. .. , s k be the elementary symmetric functions of the complex variables x 1 ,. .. , x k. We say that F $in$ C[s 1 ,. .. , s k ] is a trace function if their exists f $in$ C[z] such that F (s 1 ,. .. , s k ] = k j=1 f (x j) for all s $in$ C k. We give an explicit finite family of second order differential operators in the Weyl algebra W 2 := C[s 1 ,. .. , s k ] $partial$ $partial$s 1 ,. .. , $partial$ $partial$s k which generates the left ideal in W 2 of partial differential operators killing all trace functions. The proof uses a theorem for symmetric differential operators analogous to the usual symmetric functions theorem and the corresponding map for symbols. As a corollary, we obtain for each integer k a holonomic system which is a quotient of W 2 by an explicit left ideal whose local solutions are linear combinations of the branches of the multivalued root of the universal equation of degree k: z k + k h=1 (--1) h .s h .z k--h = 0.
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The aim of this paper is to study in details the regular holonomic $D-$module introduced in cite{[B.19]} whose local solutions outside the polar hyper-surface ${Delta(sigma).sigma_k = 0 }$ are given by the local system generated by the local branches of the multivalued function which is the root of the universal degree $k$ equation $z^k + sum_{h=1}^k (-1)^h.sigma_h.z^{k-h} = 0 $. Note that it is surprising that this regular holonomic $D-$module is given by the quotient of $D$ by a left ideal which has very simple explicit generators despite the fact it necessary encodes the analogous systems for any root of the universal degree $l$ equation for each $l leq k$. Our main result is to relate this $D-$module with the minimal extension of the irreducible local system associated to the difference of two branches of the multivalued function defined above. Then we obtain again a very simple explicit description of this minimal extension in term of the generators of its left ideal in the Weyl algebra. As an application we show how these results allow to compute the Taylor expansion of the root near $-1$ of the equation $z^k + sum_{h=-1}^k (-1)^h.sigma_h.z^{k-h} - (-1)^k = 0 $.
We consider the Weyl algebra A (=A_n(k)) and its Rees algebra B with respect to the Bernstein filtration. The homogenisation of a differential operator in A is an element in B. In this paper we establish the validity of the division theorem for homogenized differential operators and Buchbergers algorithm for computing Groebner (or standard) bases in B. As an application we describe an algorithm for computing delta-standard bases in the Weyl algebra A.
A theorem of N. Katz cite{Ka} p.45, states that an irreducible differential operator $L$ over a suitable differential field $k$, which has an isotypical decomposition over the algebraic closure of $k$, is a tensor product $L=Motimes_k N$ of an absolutely irreducible operator $M$ over $k$ and an irreducible operator $N$ over $k$ having a finite differential Galois group. Using the existence of the tensor decomposition $L=Motimes N$, an algorithm is given in cite{C-W}, which computes an absolutely irreducible factor $F$ of $L$ over a finite extension of $k$. Here, an algorithmic approach to finding $M$ and $N$ is given, based on the knowledge of $F$. This involves a subtle descent problem for differential operators which can be solved for explicit differential fields $k$ which are $C_1$-fields.
We explain that in the study of the asymptotic expansion at the origin of a period integral like $gamma$z $omega$/df or of a hermitian period like f =s $rho$.$omega$/df $land$ $omega$ /df the computation of the Bernstein polynomial of the fresco (filtered differential equation) associated to the pair of germs (f, $omega$) gives a better control than the computation of the Bernstein polynomial of the full Brieskorn module of the germ of f at the origin. Moreover, it is easier to compute as it has a better functoriality and smaller degree. We illustrate this in the case where f $in$ C[x 0 ,. .. , x n ] has n + 2 monomials and is not quasi-homogeneous, by giving an explicite simple algorithm to produce a multiple of the Bernstein polynomial when $omega$ is a monomial holomorphic volume form. Several concrete examples are given.
We prove Carleson embeddings for Bergman spaces of tube domains over symmetric cones, we apply them to characterize symbols of bounded Ces`aro-type operators from weighted Bergman spaces to weighted Besov spaces. We also obtain Schatten class criteria of Toeplitz operators and Ces`aro-type operators on weighted Hilbert-Bergman spaces.
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