# Identity of J. L. Rabinowitsch (of Rabinowitsch Trick)

For some time, it seemed widely accepted that G. Y. Rainich was the author of the note Rabinowitsch, J. L., Zum Hilbertschen Nullstellensatz., Math. Ann. 102, 520 (1929). JFM 55.0103.04., which describes a short proof of Hilbert's Nullstellensatz by what became known as Rabinowitsch Trick. This claim appears to be solely based on Bruce Palka's Editor's Endnotes", p. 460 of Am. Math. Monthly 111, No. 5 (May, 2004), p. 460, who reports:

Peter May shares the following correspondence that he received from Richard Swan. [..] The following anecdote may explay why you couldn't find him [Rabinowitsch]. Unfortunately I can't remember who told me this.

It seems that Rainich was giving a lecture in which he made use of a clever trick which he had discovered. Someone in the audience indignantly interrupted him pointing out that this was the famous Rabinowitsch trick and berating Rainich for claiming to have discovered it. Without a word Rainich turned to the blackboard, picked up the chalk, and wrote RABINOWITSCH

He then put down the chalk, picked up an eraser and began erasing letters. When he was done what remained was

               RA IN  I CH


He then went on with his lecture.

Surprisingly, this anecdote without proper source has become universally accepted knowledge, making it, e.g., into Wikipedia, or the MO thread on mathematician's pseudonyms. In several comments, KConrad (some as recent as Feb 11, 2022 - many thanks to Manfred Lehn for notifying me about this) questions this attribution, observing that

1. Rainich's work was mainly in general relativity, rather unrelated to algebra.
2. Rainich changed his name from Rabinovich upon arrival in the U.S. in 1922, and is unlikely to publish under his old name with affiliation in Moscow (where he never lived or worked).
3. The initials J.L. don't fit to Rainich/Rabinovich's G. Y./Y. G. initials.
4. There are several versions of this story before 1929 (naturally, without mentioning of the trick).

For the attribution in zbMATH, I looked further into this issue, and I think it is worth a separate thread.

The topic argument is likely less relevant - his main field until 1922 was algebraic number theory, including G. Rabinowitsch, Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern., J. für Math. 142, 153-164 (1913). JFM 44.0243.03. which earned him a talk of 1912 ICM G. Rabinowitsch, Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern., Proc. 5. Intern. Math. Congr. 1, 418-421 (1913). JFM 44.0244.01. (more about this later). Clearly, he had sufficient algebraic expertise.

However, all other objections are valid. I checked the biographies at the Faculty history of U Mich, Inna Rikun's Russian Biography and Notes on Jewish History and "Orbiting in Eccentric Circles: A Family History" by Rainich's daughter Alice Rainich Nichols.

First, one notes that none of them mentions the Rabinowitsch Trick, least of all its attribution to Rainich. All make it clear that Rainich didn't use his old name after fleeing to the U.S. in early 1923. The reasons for his escape from Odessa (he was arrested in 1922 - some sources say for unknown reasons, while his daughter claims that his work in General Relativity was incompatible with then Communist Doctrine, so he was declared enemy of the state - and nearly died in prison) make it highly unlikely to claim an affiliation in Moscow (actually, he published just in the preceding volume of the Annalen his

G. Y. Rainich, Über die analytische Funktion auf einer Minimalfläche., Math. Ann. 101, 386-393 (1929). JFM 55.0400.05. as G. Y. Rainich in Ann Arbor, Mich. U. S. A.)

Though his change in first names was somewhat unusual (from Yuri Germanovich to George Yuri, which is basically a duplication since Yuri is the Slavic George, to the effect of switched initials, and dropping his father's name Germanovich), the initial L. appears nowhere else in his work (some caution is required here since the paper is often quoted with wrong initials, e.g., in the Am. Math. Monthly anecdote).

Perhaps the strongest indication comes from the versions of the anecdote. Mordell's 1971 Reminiscences give the original:

In 1923, I attended a meeting of the American Mathematical Society held at Vassar College in New York State. Some one called George Yuri Rainich from the University of Michigan at Ann Arbor, gave a talk upon the class number of quadratic fields, a subject in which I was then very much interested. I noticed that he made no reference to a rather pretty paper written by one Rabinowitz from Odessa and published in Crelle's journal. I commented upon this. He blushed and stammered and said, "I am Rabinowitz." He had moved to the U.S.A. and changed his name. This story is known all over the U.S.A. Occasionally some one from Ann Arbor dines at John's and I ask them if they know Rainich. Yes, they say, there is a funny story about him. "Stop," I say, "let me tell you the story."

This clearly refers to his 1913 work cited above, not the Nullstellensatz. Apparently, this anecdote enjoyed wide circulation (Mordell himself points this out). In his daughter version from her book (pp. 30-31):

Yuri Germanovich Rabinovich and George Yuri Rainich

In the American world of mathematics, Yuri Germanovich Rabinovich and George Yuri Rainich came together at Vassar University in 1925, when Professor L J. Mordell challenged the originality of a paper given by my father. Upon immigrating Yuri, in order to simplify, changed his surname to Rainich by omitting the third, sixth and seventh letters of Rabinovich (he had hoped to make the change by omitting the third, fifth and seventh, as consecutive primes, but it didn't work out), and had also reversed the initials of his first and his middle name (patronymic), thus becoming G. Y. Rainich, the name by which he was known in America,

Yuri had just presented a paper on 'Expansion of Simple Factors in Quadratic Fields', when Mordell commented: "Perhaps you are unaware, Professor Rainich that this work was already done by Rabinovich, in Odessa." "But I am Rabinovich!" Yuri replied. This was the origin of the story that has followed him, as well as me, throughout our lives. This incident is part of American mathematical history. Yuri had still been Rabinovich when he presented the paper in 1912 at the International Congress of Mathematicians at Cambridge. The paper was immediately published in the very prestigious German publication: 'Zeitschrift fur Angewandte Mathematik and Physik' (Journal of Applied Mathematics and Physics), better known as 'Crelle's Journal'. Though Yuri was already a well-known scholar at the time having presented papers at Twelfth and Thirteenth meetings of 'Russian Mathematicians and Physicists,' this publication proved to play a very important role in the young scholar's life.

Eighty-seven years after this scene played out I read for the first time a different conclusion to the incident. In the newer (to me) version, Yuri, upon being challenged, walked over to the blackboard, and picking up the chalk wrote 'RABINOVICH' in large letters. He then erased the 'B', the 'O' and the 'V'. We shall never know which version is the true one.

Despite small differences (1925 instead of 1923, and the misnomer of Crelle's journal), this agrees with Mordell, and the referred second variant from 2012 has been obviously triggered by the Am. Math. Monthly variant of the anecdote. Alice Nichols doesn't notice that in the meantime, also the discussed work changed to the Trick.

A good indication how many embroidered variants circulated is the anecdote accompanying the Rainich Lecures at U Mich

Professor Rainich is a Russian by birth and an interesting story is told of an experience he had while lecturing at Columbia University. Speaking on Relativity before the faculty he quoted a number of the foremost authorities known upon the subject. One of the members of the Columbia faculty spoke up saying, "That is all very well but why don't you quote what Rabinovich has said upon this subject? "Professor Rainich was somewhat embarrassed as he replied, "Well, you see, I am Rabinovich."

Here, already, the subject has been made up to fit better into Rainich's later main field (and Vassar College has been changed to Columbia). Since Rainich's only work on General Relativity under the name of Rabinovich was a 1923 Russian translation of Eddington's "Space, time and gravitation", this version is obviously fabricated. Given the reported wide circulation, it was perhaps inevitable that a version including the Rabinowitsch trick would eventually come up.

For all these reasons, I think we can safely conclude that there is no proof of attributing the Rabinowitsch Trick to Rainich (the paper is now excluded from his zbMATH author profile), and it is indeed highly likely it was discovered by another person. So the main question is, by whom?

Nineteen years later, there is a Yu. L. Rabinovich in the records, working in analysis and, according to Rabinovich, Yu. L.; Nesterov, S. V., General form of linear differential equations whose order is lowered by means of a generalized differentiation operator (D^ \alpha), Sov. Math., Dokl. 2, 476-479 (1961); translation from Dokl. Akad. Nauk SSSR 137, 1309-1311 (1961). ZBL 0102.07302., working at Moscow State University in 1961. Though such publication gaps during the time of terror and war are not uncommon, there is, however, no clear link to the J. L. Rabinowitsch from 1929 Nullstellensatz.

Question 1. Are there any sources about the biography of Yu. L. Rabinovich, who published 1948-1961 on analysis (at least partially based in Moscow), that may confirm or rule out a link to the author of Rabinowitsch's Trick? (He did also three reviews for Math Reviews 1948-1963).

Perhaps the closest link, however, is to the author of "Sur les courbes planes du quatrième ordre possédant deux points doubles", Mathesis 45, 286-290 (1931). JFM 57.0827.02, a M. Rabinowitch from Liège. It's not just the next chronological appearance of this name in the math literature, it deals also with a topic closely related to the note in Mathematische Annalen two years earlier - indeed, one may easily imagine how studying hyperplane sections of singular varieties in the singular points may lead to the Rabinowitsch Trick.

So, as a working hypothesis, the author of the Mathesis paper appears to be a natural candidate to identify with the inventor of the Rabinowitsch Trick, which leads to

Question 2. Is there any information available on a mathematician named Rabinowitch who worked in Liège around 1931, and can it be confirmed (or ruled out) that he moved there from Moscow shortly before?

And, of course, more general:

Question 3. Are there any other sources which may contribute biographical information about the author of the Rabinowitsch Trick?

• In the title, would it not make sense to use the spelling "Rabinowitsch" for the person in question? Feb 19 at 23:37
• A different list of papers by Yu. L. Rabinowitsch is on the mathnet.ru site: mathnet.ru/php/…. It has some overlap with the zbmath link in your post at his name, but neither list is a subset of the other. All the mathnet.ru papers are related to differential or integral equations. Feb 20 at 4:35
• For what it's worth, the Russian version of the page en.wikipedia.org/wiki/Dmitrij_Kostomarov indicates that person was one of the students of Yu. L. Rabinovich. Feb 20 at 4:41
• Many thanks - indeed, title corrected. Feb 20 at 4:44
• Indeed - actually, we added the mathnetru-ID to the zbMATH profile in the course of the thread, but it will only be visible after the next update. Additionally, I found that the Russian translation of Courant's "Differential and Integral Calculus" [Moscow, Nauka (1967, zbmath.org/0153.08401)] was also done by Yu. L. Rabinovich ikfia.ysn.ru/wp-content/uploads/2018/01/Kurant_t1_1967ru-1.pdf Feb 20 at 4:58