[Date Index] [Thread Index] [FOM Postings] [FOM Home]

*To*: fom@math.psu.edu*Subject*: FOM: Chudnovsky's contribution to MRDP*From*: martind@cs.berkeley.edu (Martin Davis)*Date*: Tue, 10 Feb 1998 14:17:28 -0800*Sender*: owner-fom@math.psu.edu

Joe Shipman has requested that I tell what I know about this. It's a long murky tale, and I'm not sure how relevant it is to F.O.M., but on inquiry, Steve encouraged me to go ahead. THE POLITICAL BACKGROUND: In 1969, Yuri Matiyasevich, 22, was in Leningrad, and Gregory Chudnovsky, 17, was in Kiev. Chudnovsky is Jewish and Matiyasevich is not. This is relevant because Soviet mathematics was riddled with anti-semitism, and the Chudnovsky's were later to claim that the denial of credit to Gregory was part of this pattern. THE MATHEMATICAL BACKGROUND: In our 1961 paper, Hilary Putnam, Julia Robinson and I proved the implication: JR ==> Every r.e. set is Diophantine where JR is Julia's hypothesis that there is a Diophantine function f which is O(x^x) but not O(x^k) for any fixed k. (I never miss an opportunity to note that Kreisel said in his MR review of this paper "... it is likely the present result is not closely connected with Hilbert's tenth problem" while omitting to inform his readers of the above implication.) A 1968 paper of mine is particularly relevant to Chudnovsky's claim. In that paper, I introduced the equation 9(x^2+7y^2)^2 - 7(u^2+7v^2)^2 = 2; I deduced JR from the assumption that that equation had no solutions other than the trivial x=u=1,y=v=0. In fact that equation has many solutions, but my proof can easily be modified to show that JR would follow if there are only finitely many solutions. [That question remains open and not without interest; if true, it would follow that the Turing degree of the set of Diophantine equations with infinitely many solutions is exactly 0".] WHAT YURI MATIYASEVICH DID IN 1970: Yuri gave a (beautiful) Diophantine definition of the function F_{2n} where F_n is the n-th Fibonacci number, thereby proving JR. WHAT GREGORY CHUDNOVSKY PUBLISHED ON THE QUESTION: I know of two relevant publications. One is a published paper that uses methods very similar to Yuri's to prove JR, but referring to the solutions of the Pell equation (x^2 - dy^2 = 1) instead of the Fibonacci numbers. Other proofs of JR using the Pell equation were also given shortly after Yuri's paper, using Yuri's methods, by Kosovski in Russia and by me (among others). Later work by Yuri and Julia reducing the total number of unknowns needed in Diophantine definitions of r.e. sets used the Pell equation. Gregory's other paper was a crudely mimeographed private publication done in Kiev. It announced a proof of JR based on my 4th degree equation (mentioned above), but furnished no details. (See a quotation from this publication in Matiyasevich's book "HILBERT'S TENTH PROBLEM" MIT 1993, p.39; this book has a very complete bibliography.) WHAT JULIA TOLD ME OF HER CONVERSATIONS WITH YURI: Julia spent some time in Leningrad working with Yuri. She was very concerned about anti-semitism in Soviet mathematics, not only in connection with Hilbert's tenth problem. She questioned Yuri closely about Gregory's contribution among other matters. Yuri claimed that Gregory's published paper was done after Gregory knew his work, that Gregory had attended a lecture in Kiev given by Yuri, and that he was entitled to no more credit for his Pell equation proof of JR than Kosovski or I were for ours. Finally, he stated that although Gregory's original claim was based on my 4th degree equation, his published proof used Yuri's methods. MY CONVERSATION WITH THE CHUDNOVSKY'S: I talked to the Chudnovsky brothers at the ICM in Helsinki 1978. David did most of the talking. Their story was that Markov had both proofs (Yuri's and Gregory's) in hand, but simply decided that "there will be only one solver of Hilbert's tenth problem" and threw Gregory's out. I mentioned my 4th degree equation to Gregory and explained why it was still important. I asked whether he had indeed proved that its number of solutions is finite. He didn't answer my question but said that that had been long ago and he was currently interested in other matters. THE "NEW YORKER" PROFILE: In March 1992, The New Yorker published a "Profile" article on the Chudnovsky brothers. The article was mainly about their home-made "super-computer" for which they were seeking financial support and which they were using to churn out large numbers of decimal digits of \pi, seeking to discern a pattern. The article briefly mentioned Gregory's work on Hilbert's tenth problem and quoted the brothers as saying "Matyasevich [sic] has recently said that the Chudnovsky method is the preferred way to solve Hilbert's Tenth Problem". I sent Yuri email calling this to his attention. He indicated that he personally did not intend to respond, but suggested my writing the magazine asking on what basis the statement was being made. I did so in a letter briefly explaining my own involvement with the area and explaining that I wrote at Yuri's suggestion. I received no reply either to this letter or to a follow-up letter I wrote a month later. THE TRUTH: Alas! History is not mathematics. With all of this, I still don't know what "really" happened in Kiev and Moscow in 1970 and don't imagine that I ever will. But on the basis of what I do know, I see no reason to regard the proof of JR as due to anyone but Yuri. Martin

[Date Prev] [Date Next] [Thread Prev] [Thread Next]

[Date Index] [Thread Index] [FOM Postings] [FOM Home]