LECTURE NOTES

ON HEURISTIC GENERALIZATIONS OF THE JACOBIAN CONJECTURE

By Pascal Kossivi ADJAMAGBO

University Paris 6 – Institut de Mathématiques de Jussieu-UMR 7586 CNRS

Opening lecture of the International Conference at Chern Institute of Mathematics on

Affine Algebraic Geometry and the Jacobian Conjecture” on 21-25 July 2014

Tianjin, 21 July 2014

Abstract

The aim of this opening lecture is to unveil heuristic and stimulating generalizations of the Jacobians Conjecture. By lack of time, we will consider only the first level of such generalisations, concerning first polynomial endomorphisms of the algebraic affine space in any dimension and in any characteristic, secondly complex analytic endomorphisms of the complex affine space of any dimension, thirthly real analytic endomorphisms of the real affine space of any dimension, and finally real algebraic endomorphism of the real affine space of any dimension. The complete levels of generalisation could be quiety read in the complete written version of this lecture. These restrictions of the anounced three levels of generalizations will be enough to unveil the surprising and hidden vitality and fecundity in many branches of mathematics of the venerable 75 years old Jacobian Conjecture to whom we are paying homage during these memorable days in the beating heart of the more venerable multi-millennary alive chinese civilisation and empire.

0. Introduction

Let me begin by drawing your attention on some points of the history and the epistemology of mathematics which are not well known, to apply to mathematics the African proverb claiming that “if you don’t know were you are going to, try to know were you are coming from”.

Indeed, at the beginning of a mathematical papyrus dated from 1650 BC and coming from another venerable multi-millenary civilisation, the Antic African Egyptian dead civilisation, its authors, the oldest known mathematician of history, calling himself Ahemesu, and not Ahmes as unproperly usually pronounced, presented the contents of the document by writing : “exact methods of investigation of Nature, in order to know every thing which exists but is hidden”. This mention which is considered by some historians of mathematics as the title of the presented document, could more rightly be understood as a comment or reflexion on mathematics, more precisely on the essence of mathematics essentially consisting in exactness and rigor, on the nature of mathematics consisting in methods, and on the finality of mathematics consisting in the knowledge of the conceptual and physical components of the Nature. So according to me, this reflexion of Ahemesu is the deepest, the most pertinent, fecund and genius thought on mathematics ever enounced by the human mind.

Let us now explain the relation between Ahemesu statement and the Jacoian Conjecture, by unveiling the light that Ahemesu statement project on the Jacobian Conjecture, more precisely on the fecund heuristic approaches of this conjecture. Indeed, in accordance with Ahemesu statement, we could first guess, then believe, that behind the Jacobian Conjecture, there exists a mathematical law, or a law of the Nature as would Ahemesu say, which governs the conjectured jacobian phenomena, and which should be discovered by appropriate exacts, heuristical and fecund methods.

In accordince with Ahemesu statement, we could believe that this law which is hidden behind the Jacobian Conjecture, and which does not does not run away as a mathematical explorer approches, is the law of “passage from the local to the global injectivity” in different branches of mathematics, more precisely, not only in complex algebraic geometry as in the historical statement of the Jacobian Conjecture, but also in algebraic geometry over fields of any characteristic, in particular over finite fields which are more “real” for computer than the fields of “real or complex” numbers, in complex analytic geometry, in real analytic geometry, and finally in real algebraic geometry.

In accordance with Ahemesu statement, we do believe that appropriate “exact” methods in these various areas of mathematics will lead to the proof the various appropriate heuristic generalizations of the Jacobian Conjecture, and that these proofs should be inspired by the evocated hidden mathematical law of “passage from local to global injectivity”.

So let us expose now more precisely the first level of the anounced “heuristic generalizations of the Jacobian Conjecture”, after recalling the “local and global injectivity” reformulation of the historical Jacobian Conjecture.

1. The local and global injectivity formulation of the Complex Jacobian Conjecture

1.1 Remark

According to Osgood Theomem (see for instance (8) Rosay for a clever proof), claiming that any injective analytic map from an open subset of a complex affine space into itself is an “open embedding”, i.e. its image is open, the map induces an isomorphism of complex analytic manifold from its definition domain and its image, the historical Complex Jacobian Conjecture, proposed by O.-H. Keller in 1939 (6), could be reformulated as follows :

1.2 Conjecture

Any locally injective polynomial map from a complex affine space to itself is not only injective, but also bijective with a polynomial inverse.

1.3 Remark

1) We do believe that it is this formulation, forgetting the jacobian condition, which expresses the deepest meaning of the historical Complex Jacobian Conjecture, and which is the most geometric, fecund and heuristic formulation of this conjecture.

2) The historical and epistemological interest of this conjecture has been strenghtened since 2005 by the proof first by Y. Tsuchimoto (10), then by A. Belov and M. Kontsevich (4) of its equivalence with the Dixmier Conjecture (5), claiming that “any endomorphism of any complex algebra of differential operators with polynomial coefficients is an automorphism”, and by the proof by A. van den Essen and myself (2) of its equivalence also with the Poisson Conjecture, claiming that “any endomorphism of any complex Poisson algebra of polynomials is an automorphism”.

2. The Jacobian Conjecture in any characteristic

2.1 Remark

In their famous 1982 paper (3), H. Bass, E. Connell, and D. Wright pointed out a trivial counter-example to the analogous of the Complex Jacobian Conjecture in dimension one for fields of positive characteristic, by considering the polynomial X – X^p, whose derivative is 1 over any field of characteristic p, whose fiber above zero contains the whole sub prime field of characteristic p according to the “Little Fermat Theorem”. Illustrating a sentence of the philosopher Hegel claiming that “Falsness is a moment of Truth”, we proposed in our lecture at 1995 Curacao Conference on the Jacobian Conjecture (1) that for fields of positive characteristic, the statement of the Complex Jacobian Conjecture should be rectified just by adding that the geometric degree of the considered map is not a multiple of the characteristic, and by giving the following precise meaning to “local injectivity”, knowing that it can be proved that this meaning is equivalent to the “unramifiedness” of the considered morphism.

2.2 Definition

A morphism of an algebraic variety over an algebraically closed field is said to be “locally injective” if it induces a “closed embedding” from the spectrum of the completion of the local ring of any closed point of the domain of definition of the morphism to the corresponding completion for the image of this point by the morphism.

2.3 Conjecture

Any locally injective polynomial map from an affine space over an algebracally closed field to itself, with a geometric degree not divisible by the characteristic of the field, is not only injective, but also bijective with a polynomial inverse.

3. The Complex Analytic Jacobian Conjecture

3.1 Remark

In their quoted paper (3), H. Bass, E. Connell and D. Wright also pointed out a non trivial counter-example to the analogous of the Complex Jacobian Conjecture for complex analytic endomorphisms of of a complex affine space, with not finite fibers. Illustrating again the sentence of Hegel, we propose the following Complex Analytic Jacobian Conjecture :

3.2 Conjecture

Any locally injective complex analytic map from a complex affine space to itself, with finite fibers, is not only injective, but also an open embedding.

4. The Real Analytic Jacobian Conjecture

4.1 Remark

In 1994 Pinchuk (9) pointed out a polynomial endomorphism of the real affine plane, which is a local diffeomorphism without being neither injective, nor surjective, with a image equal the real plane deprived with two points, hence which is not simply connected. So this map is an irrefutable counter-example to the naïve real generalization of the Jacobian Conjecture. Illustrating again Hegel sentence, let us rectify and generalize to real analytic endomorphisms as follows this naïve real generalization of the Complex Jacobian Conjecture :

4.2 Conjecture

Any real analytic map from a real affine space to itself with finite fibers, which is a local diffeomorphism with a simply connected image, is not only injective, but also an open analytic embedding.

5. The Real Algebraic Jacobian Conjecture

5.1 Remark

According to the remarkable surjectivity of injective continous maps with an algebraic graph from a real affine space to itself proved by Kurdyka and Rusek in 1988 (7), the last conjecture could be refined as follows :

5.2 Conjecture

Any Nash map, i.e. a real analytic map with an algebraic graph, from a real affine space to itself, which is a local diffeomorphism with a simply connected image, is not only injective, but also an analytic diffeomorphism.

References

(1) K. Adjamagbo, On separable algebras over a U.F.D. and the Jacobian Conjecture in any characteristic, in Automorphisms of Affine Spaces, Ed. Van den Essen, 89-103, 1995, Kluwer Academic Publishers, Proceedings of the Conference on “Inertible Polynomial Maps”, Curaçao, July 4-8 1994

(2) P. K. Adjamagbo, A. van den Essen, A proof of the equivalence of Dixmier, Jacobian and Poisson conjectures, Acta Math. Vietnamica, Vol. 32, Number 2-3, 2007, 205-214

(3) H. Bass, E. Connell, D. Wright, The Jacobian Conjecture : Reduction of Degree and Forma Expansion of the Inverse, Bull. A.M.S. 7 (1982), 287-330

(4) A. Belov, M. Kontsevich, Jacobian Conjecture is stably equivalent to Dixmier Conjecture, Mosc. Math. J., 2007,Volume 7, Number 2, Pages 209–218

(5) J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France, 96 (1968), 209-242

(6) O.-H. Keller, Ganze Cremona-Transformation, Monats. Math. Physik 47(1939), 1, 299-306

(7) K. Kurdyka, K. Russek, Surjectivity of certain injective semialgebraic transformation of R^n, Math. Z. 200 (1988), 141-148

(8) J.P. Rosay, Injective endomorphisms of algebraic varieties, Amer. Math. Monthly. 89 (1982), 587-588

(9) S. Pinchuk, A counterexample to the real Jacobian Conjecture, Math. Zeitschrift, 217 (1994), 1-4

(10) Y. Tsuchimoto, Endomorphisms of Weyl Algebra and p-curvatures, Osaka J. Math., 42(2005), 435-452

 

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