A. van den Essen TO BELIEVE OR NOT TO BELIEVE: THE JACOBIAN CONJECTURE

In this paper we try to convince the leader that there is no good reason to believe that the Jacobian Conjecture holds. Although there are several arguments in favor of this conjecture, we show that these arguments haven't got the power to justify the statement that the Jacobian Conjecture holds in general. Let F : C n —> C n be a polynomial map i.e. a map of the form {xiì...,xn) i-> (F1(xiì...,xn),...,Fn(x1,...,xn)) where each Fi belongs to the polynomial ring C[X] := C[X\ ,Xn]. Linear maps are examples of polynomial maps, namely with Fi(xiì...ìxn) = anxi H V ainxn, a „ G C forali ij. From linear algebra we know (for linear maps): 1) F is bijective if and only if F is injective. Furthermore the inverse is again a linear map. 2) F is invertible if and only if det(a i i) e C* (= C\{0}). Now the question is: can we extend these rèsults to polynomial maps? The answer to the first question is YES. More precisely, the bijectivity was first proved in [3] and the first proof of the following result was given in [6]. T H E O R E M . Let F :C -> C n be a polynomial map. If F : € -» C n is injective, then F is bijective and furthermore the inverse is again a polynomial map! So now let us look at the second question. Suppose that the polynomial map F : C n —> C is invertible. Then by the locai inverse function theorem at each point z e C we get det JF(z) ^ 0. So the polynomial det JF does not vanish on C n . So det JF e C*. So we get 284 A. van den Esseri PROPOSITION. Let F ':. C -> C be a polynomial map. If F ìs invertible, then det JF e C*. In [25] Keller asked: is the converse true? This problem is now known as Keller's problem but more as Jacobian Conjecture JC(n). If F : C —> C is a polynomial map with de£ J C € C* (we cali such maps fó//er maps), then F is invertible (or equivalently F injectìve). Since that time many people have worked on this problem and several false proofs were published. Every year I stili receive many new false proofs. In spite of ali these efforts thè conjecture is stili open for ali n > 2!! (We refer the reader to the following survey papers [2], [11], [13], [14] and [34].) What I like to do in this lecture is to describe you some evidence in favor of the J.C. (which I denote by P.) and some evidence against it (C). It is up to you to BElieve or not to BElieve the Conjecture. P. Let me start describing that there is an overwhelming evidence for the 2 dim J.C. Let F = (/, g) : C -> C be a Keller map. a) It was proved by Moli in [31] that F is invertible if deg / , deg g < 100. b) Appelgate, Onishi ([1]): if deg / or deg g is a product of at most 2 prime numbers, then F is invertible. e) Gwozdziewics ([22]): if there exists one line / C C such that F\i : l -* C is injective, then F is invertible. C. Let me comment on these results: Consider the n-th cydolomie polynomial $n{X) := FI ( ^ "~ C) (-Pn = t n e s e t of primitive n-th roots of unity in G). Then it is well-known that $ n (^0 ^ Z[X]. Doing some calculations for some values of n one observes that ali coefficients of $n(X) belong to {0, —1,1}. In fact one can prove a) if n < 100 then ali coefficients of $n(X) are in {0, —1,1}. b) If n is a product of at most 2 prime numbers, then ali coefficients of $n{X) belong to {0,-1,1}. So we are in a similar situation as above. However, if n = 105 = 3.5.7 the coefficient of X is -2! ! P. Let me give more evidence in case n > 2. In [2] Bass, Connell and Wright and independently in [36], Yagzhev showed THEOREM. Iffor ali n>2 and ali Keller maps F : C • C with deg Fi < 3 for ali i F is invertible, then J.C. is true!