Using the Alpoge-Claude amazing counterexample of the Jacobian Conjecture to Generate Many More of Them

By Shalosh B. Ekhad and Doron Zeilberger

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First Written: Aug. 3, 2026

This version: Aug. 3, 2026

Maple package

Sample Input and Output for Alpoge.txt

  • If you want to see the five-parameter counterexample, inspired by the Alpoge-Claude example, by "cheating" and looking at the monomials that showed up, and setting a generic transormation with the same monomials but arbitrary coefficients

    then the input gives the output.

    [Note that this is hard-wired in the Maple packge Alpoge.txt by procedure GenClaude(x,k). Also note that it took less than a second!]

  • But that was "cheating" since we used the same template. For a still restricted, but much larger haystack, still with total degrees 7,6,4

    then the input gives the output.

    Note that we got the same output, nothing new!

  • If you want to see all the counterexamples where the largest degree is 6 (with this restricted choice of supports)

    then the input gives the output.

    [Note that it is the empty set, nothing was found. Of course, it is still possible that there are degree 6 counterexmaples with larger supports.]

  • If you want to see a more exhaustive search for transoformations of degree 7 (but still not fully general)

    then the input gives the output.

    [Note the nothing new was found]

  • If you want to see the analogous thing for degree 8

    then the input gives the output.

    [Note: it seems that we found a new one! Alas the degree ≤ 8 turned out to be degree 6, so the new one is equivalent (via a change of parameters) to GenClaude(x,k)]

  • If you want to see the truncation to total degree 20, of the formal power series inverse transformation of Claude(x)

    then the input gives the output.

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