Andy's Math/CS page

Wednesday, July 21, 2010

Injective polynomials

From a paper of MIT's Bjorn Poonen, I learned of an amazingly simple open problem. I'll just quote the paper (here Q denotes the rational numbers):

"Harvey Friedman asked whether there exists a polynomial f(x, y) in Q(x, y) such that the induced map Q x Q --> Q is injective. Heuristics suggest that most sufficiently complicated polynomials should do the trick. Don Zagier has speculated that a polynomial as simple as x^7 + 3y^7 might be an example. But it seems very difficult to prove that any polynomial works. Both Friedman's question and Zagier's speculation are at least a decade old... but it seems that there has been essentially no progress on the question so far."

Poonen shows that a certain other, more widely-studied hypothesis implies that such a polynomial exists. Of course, such a polynomial does not exist if we replace Q by R, the reals. In fact any injection R x R --> R must be (very) discontinuous.

Suppose an injective polynomial f could be identified, answering Friedman's question; it might then be interesting to look at `recovery procedures' to produce x, y given f(x, y). We can't hope for x, y to be determined as polynomials in f(x, y), but maybe an explicit, fast-converging power series or some similar recipe could be found.

Finally, all of this should be compared with the study of injective polynomials from the Euclidean plane to itself; this is the subject of the famous Jacobian Conjecture. See Dick Lipton's excellent post for more information.