An Amazing Paper
An amazing new set of conjectures and the jacobian conjecture
Arno van den Essen is one of the world experts on the Jacobian Conjecture. He has written a book on the problem—really the book. Besides this he has made many contributions to the understanding of this great conjecture.
Today I plan to talk about a paper Essen just put into the archives on the Jacobian Conjecture (JC). His paper is amazing—really amazing. Its title is The Amazing Image Conjecture.
Years ago Essen, Evangelos Markakis, and I wrote a simple paper on an approach to the JC. The paper is I still believe a viable approach to the problem. But we could only prove a very weak result about the JC problem.
Since we could not make our ideas work, lets turn to Essen’s new paper.
I thought you might like to go through a possible scenario of how the JC was created, and see some of the rationale behind the conjecture. This is not a “real” history, but one that I hope helps expose the ideas that go into the creation of a great conjecture.
Imagine that you are studying mappings from the two dimensional Euclidean plane to itself. A natural question is: when is one-to-one? There are many properties that mappings may or may not have, but being one-to-one is a basic property. Such functions are called injective—this name is due to Nicolas Bourbaki.
The first thing you notice is that being injective is a “global” property, since is one-to-one, or injective, provided there are no two distinct points and in the plane so that
Determining whether or not there are such distinct points is equivalent to asking for the solvability to the two equations
where and .
In general deciding whether such equations have a solution can be very difficult. In order to have any hope of solving this problem you decide to restrict the function to be a polynomial mapping: this means that and are both polynomials.
Even with the restriction to polynomial maps the difficulty is that and can be anywhere in the plane; they need not be “near” each other. This global nature is the key reason that determining whether or not is injective is hard.
You realize one way to simplify the problem is to ask a much weaker question: is the mapping “locally injective?” The mapping is locally injective if for each point there a disk around so that for all and in the disk if , then .
There are two reasons you think this is a good move. First, the mapping cannot be injective unless it is locally injective at every point . This is a simple idea, but a powerful one. When facing a tough question, often relaxing the question can yield insight.
Second, you recall a classic theorem that has a sufficient condition for a smooth map to be locally injective at a point. The theorem is the famous Inverse Function Theorem. In one dimension it says that a function is locally injective at point , provided
For example, is locally injective at any point .
The Inverse Function Theorem generalizes this simple idea from calculus-101 to two dimensions and beyond. The critical question is what takes the role of the derivative of a function? It turns out the role is played by the Jacobian matrix of the mapping —clearly we are close to why the conjecture is called the JC.
Associated with any mapping smooth is the Jacobian matrix defined as
Note, is the partial derivative of the function . In the one dimension case this would degenerate to the by matrix of . If the determinant of this matrix is non-zero at the point , then by the famous inverse function theorem the map is locally injective at .
Thus to tell if the mapping is locally injective at every point you need only look at the determinant of the matrix . Note this is a polynomial in the variables and . It can be a high degree polynomial, and over the reals it could be quite difficult to tell if this polynomial is or is not ever . For example, does
have a solution over the reals?
Now you make a move that is brilliant. Rather than try to figure out whether such polynomials have zeroes or not you realize that the polynomial where is a non-zero constant has no zeroes. So you make the `simplifying” conjecture:
Conjecture: If has where is a non-zero constant, then is globally injective.
This is the Jacobian Conjecture due to Ott-Heinrich Keller in 1939, and later Shreeram Abhyankar gave it the modern name.
You might ask why not replace the condition is a non-zero constant by the condition is never zero for any real numbers. The answer is surprising—it is false. This was the so called “Real Jacobian Conjecture,” which is now known to be false. See this for a counter-example due to Sergey Pinchuk.
I will not go into more details, but the intuition I believe is that many real analysis problems are only understood when complex numbers are allowed. This happens in understanding the singularities of rational functions. For example the reason the power series of has radius of convergence only is there is a complex singularity. In the reals this function has no poles, while over the complex numbers it has a pole at .
There have been many claims of proofs of JC, some are still active claims. The incorrect proofs of JC—I think all have been proofs not counter-examples—have been by amateurs and professionals alike. The problem is slippery. There seems to be many difficulties in understanding the problem. Yet the problem’s statement only requires basic calculus.
JC seems to be one of those problems that should be resolvable. Yet so far it has withstood the attacks of mathematicians for over 70 years. Since then there have been many partial results and reductions, but no proof yet.
Equivalent Problems: Like many problems in mathematics there are numerous equivalent statements. See Essen’s book for many examples. One of the most famous is to reduce the degree of mapping . In order to do this the problem is generalized to higher dimensions: the map is allowed to be
where . Hyman Bass, Edwin Connell, and David Wright proved the famous:
Theorem: If the generalized JC is true for all and polynomial maps of degree at most three, then JC is true.
As often is the case, two is special. The general JC is easily proved for mappings of degree at most two.
Direct Proofs Attempts: One “obvious” idea is to try and prove an inductive statement. If is the mapping, then prove that there is a transformation of that preserves injectivity and lowers the degree. A claim a few years ago was based roughly on this type of approach, but it fell apart.
Advanced Attempts: There were approaches to the JC problem based on advanced ideas of algebra. I am the last one to discuss these, but while they look possible they have not yet worked.
Algebra or Analysis: There are other approaches based on analysis rather than algebra. Of course all methods are allowed, but I have often thought the problem is closer to number theory than algebra, again who knows.
Essen’s amazing paper describes a beautiful connection recently discovered by Wenhua Zhao between the JC and some striking questions that seem interesting on their own. There are a new set of extremely interesting conjectures. I will explain one of them—see Essen’s well written paper for the full set of conjectures.
The generic class of open problems concern particular linear functionals on spaces of polynomials. Call “nice” if the only polynomial such that for all is the zero polynomial. The amazing fact is that there exist particular functionals such that: if is nice, then JC is true. These functionals are still fairly complicated to define, so here I will follow Essen’s paper by starting with three simpler functionals for which niceness is non-trivial—and still open for the third:
Each of sends a polynomial to a scalar. The questions are:
- If for all integers , must ?
- If for all integers , must ?
- If for all integers , must ?
Essen can prove the first two are true for univariate polynomials, but the third about is open for multi-variate polynomials.
I will give a bit of flavor of how he proves such theorems, but as always see his paper for the details.
Consider the polynomial
where and each coefficient is a Gaussian integer. I am assuming that the constant term is zero, and plan on showing the lowest term is also zero. This will inductively prove that is zero.
The key is to raise the polynomial to a special power , which is selected so all the terms but the last “disappear” when is applied. Let where is a prime and divides . Then, is congruent to
modulo . Define to be the minimum degree of the terms of . Then is at least
Thus is congruent modulo to
since all the terms in have degree at least . It is easy to prove that maps
It follows that is divisible by , and that . The latter uses that is a Gaussian integer.
Of course the open question is the JC. Is it true? This problem is not a Clay problem—perhaps it should be added to replace the Poincaré, Conjecture. The problem JC is on Steve Smale’s list of open problems as problem .