Progress On The Jacobian Conjecture
More on the crypto approach to the Jacobian Conjecture
Arno van Essen is one of the world experts on the Jacobian conjecture (JC)—we have discussed his work before here. He has made many contributions to it, with my favorite being: To Believe Or Not To Believe: The Jacobian Conjecture. I like his attitude about conjectures: I think we should be more skeptical about our own. Oh well, few of my colleagues feel this way about , for example.
Today I want to update a previous discussion on the JC, and prove a new theorem.
It does not solve JC, but has two nice properties. It shows that we can improve what we discussed before. Our previous, almost trivial observation, can be made into a a full-blown theorem. We can show that a class of polynomial maps, called -maps, are injective. This is an improvement over what was previously known.
Also the same ideas can be used to prove a new statement equivalent to JC. This statement seems like it must be true, must be provable, but since it is equivalent to JC it will certainly not be trivial. This is joint work with Essen, and we are writing up a paper that will be available for discussion soon.
I noticed before that a certain important class of maps, called -maps “leak” some information—just more than a bit per variable. This was a fun, trivial observation. Yet with some modest additional insights I realized recently that such maps leak all their information: given the value of the map, there is exactly one pre-image.
In order to state the theorem I need the notion of a -map. They are cubic maps of a special form.
Definition: Let be an by integer matrix such that . Define the polynomial function from to by where each is of the form
These are named -maps after Ludwik Drużkowski. We will write this as . The mappings themselves can be defined without the condition , but this condition seems to be quite useful.
These maps are interesting since the JC is equivalent to the statement that all such maps over the complex numbers are injective, provided as usual their Jacobians are non-zero constants.
Integer D-maps Are Invertible
Theorem: Let be an by integer matrix and let be a -map such that . Then is injective.
Namely, let . We will show that all pre-images of (in ) are equivalent modulo and from that we deduce that they are equivalent modulo for all . This implies that has only one pre-image: for if and ,then for some , does not divide i.e. is not equivalent modulo , a contradiction.
The following proof is a simplification due to Arno of a draft of mine.
Lemma 1: All pre-images of are equivalent mod .
Proof: Let . Then . So . Multiplying by , using , we get that .
Lemma 1 is essentially the “trivial” observation that I made before.
Lemma: Let . Then all pre-images of are equivalent mod .
Proof: We use induction on , the case is Lemma 1. So let and let . Write , with , for all . Then by the induction hypothesis is independent of . Let be the -th row of . Then , where denotes the usual inner product. Now substitute into this equation. This gives
Since does not depend on this gives the desired result.
This is close to solving the JC. Here “close” means that we need to do the above for maps that have coefficients that can be rational. Currently, the arithmetic arguments do not quite work: the problem is potential factors of in denominators of the entries of the matrix . Can this be overcome by a more careful analysis?