An explanation for undergraduated students about why the Jacobian conjecture is hard
In fact, we do not know if the JC is hard or not, for all what we know, some smart undergraduate can simply write a formula of a degree 3 polynomial vector-function in several complex variables that will be a counter-example to this conjecture. (On the other hand, if this were a polynomial in two variables, its degree would have to be quite high: More than 100.) However, we do know that proving the JC is hard and the history of its "proofs" is littered with failed attempts by strong professional mathematicians (two of them, Segre and Grobner, are quite famous), so maybe the undergraduate student might be interested in the following (incomplete) history of these failed efforts:
- W. Engel, Ein Satz uber ganze Cremona Transformationen der Ebene, Math. Ann. 130 (1955), 11–19.
Two mistakes there were found by Vitushkin in 1973.
- B. Segre published at least three wrong proofs of the Jacobian Conjecture:
B. Segre, Corrispondenze di Mobius e Transformazioni cremoniane intere, Atti Accad. Sci. Torino: Cl. Sci. Fis. Mat. Nat. 91 (1956–57), 3–19.
Forma differenziali e loro integrali, vol. II, Docet, Roma, 1956.
Variazione continua ed omotopia in geometria algebraica, Ann. Mat. Pura Appl. 100 (1960), 149–186.
- W. Grobner, published a false proof of the Jacobian Conjecture
W. Grobner, Sopra un teorema di B. Segre, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 31 (1961), 118–122.
- S. Oda wrote a preprint containing a faulty proof of the JC:
S. Oda, The Jacobian Problem and the simply-connectedness of $A^n$ over a field $k$ of characteristic zero, preprint, Osaka University, 1980.
If your undergraduate student still does not believe that proving the JC is hard, you should ask him/her: Could you prove for me that an injective polynomial map ${\mathbb C}^n\to {\mathbb C}^n$ is also surjective and keep in mind, that this is false for holomorphic functions ${\mathbb C}^2\to {\mathbb C}^2$. If this undergraduate comes up with an ingenious proof (not based on a literature search), you might be talking to a future Fields medalist. Next, you should ask: What kind of tools do you think one can use to prove the JC for polynomial functions of several complex variables (it suffices to treat this case only). After all, this looks like a complex analysis problem, right? After your friend struggles for a few hours/days, write the following example: $$ f(z, w) = (ze^{−w}, e^w). $$ Clearly, $J_f=1$ everywhere, but this function is not injective.
Now, it might be the right time to ask "why should the JC be true at all?", since having constant Jacobian does not seem to correlate with injectivity! So, you suggest your undergraduate friend to look for a counter-example (see line 1).
I always believe an example is great to explain something abstract. So I will refer the counterexample for the strong real Jacobian conjecture, which is shown by Sergey Pinchuk.
See https://link.springer.com/article/10.1007%2FBF02571929.