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An effective approach to Picard-Vessiot theory and the Jacobian Conjecture

Bogdan P., Hajto Z., Adamus E.

Schedae Informaticae

2017

Vol. 26

49--59

In this paper we present a theorem concerning an equivalent statement of the Jacobian Conjecture in terms of Picard-Vessiot extensions. Our theorem completes the earlier work of T. Crespo and Z. Hajto which suggested an effective criterion for detecting polynomial automorphisms of affine spaces. We show a simplified criterion and give a bound on the number of wronskians determinants which we need to consider in order to check if a given polynomial mapping with non-zero constant Jacobian determinant is a polynomial automorphism. Our method is specially efficient with cubic homogeneous mappings introduced and studied in fundamental papers by H. Bass, E. Connell, D.Wright and L. Drużkowski.

Sub-Pfaffian sets and a generalization of Wilkie's theorem

Hajto Z.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 2

181-189

We prove an analytic generalization of A. Wilkie's well-known theorem on the model completeness of the theory of the real field with exponentiation.

Finite linear groups as differential Galois groups

Crespo T., Hajto Z.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 4

361-373

An effective construction of linear differential equations with Galois group a given finite group is presented.

Lemmass A and B for sub-Pfaffian sets

Hajto Z.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 4

325-336

In this paper we prove two fundamental lemmas of sub-Pfaffian geometry which are counterparts of Lemmas A and B for subanalytic sets [4]. We use a generalized version of the Tangent Mapping Theorem [2], following our program announced in [11].