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Register a new userEffective descent for differential operators
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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.
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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.
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.
We prove that $p$-determinants of a certain class of differential operators can be lifted to power series over $mathbb{Q}$. We compute these power series in terms of monodromy of the corresponding differential operators.
In this note, we will compare the Garnier system in two variables with four-dimensional partial differential system in two variables with $W(D_6^{(1)})$-symmetry. Both systems are different in each compactification in the variables $q_1,q_2$, however, has same five holomorphy conditions in the variables $p_1,p_2$.
We prove descent theorems for semiorthogonal decompositions using techniques from derived algebraic geometry. Our methods allow us to capture more general filtrations of derived categories and even marked filtrations, where one descends not only admissible subcategories but also preferred objects.
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