When Abhyankar and Moh introduced the term 'Jacobian conjecture' in the 1970s-80s, they referred only to the two-dimensional case.
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
The claim has two parts: that Shreeram Abhyankar and Tzuong-Tsieng Moh introduced the name "Jacobian conjecture", and that when they did so they meant only the two-variable problem. Both parts have real support, but of an indirect kind. Moh has himself recalled, in a 1998 lecture, that after Oscar Zariski remarked at a Purdue seminar in the late 1960s that the assertion was still unproved, "we decided to call it the Jacobian Conjecture", and Alexander Borisov has attributed the coinage to Abhyankar, which bears out the attribution of the name to Abhyankar. Everything Abhyankar and Moh published under the name in the period, including Abhyankar's 1977 Tata lectures and Moh's 1983 paper, concerns the plane, which is the two-variable focus of their own work, and the current Wikipedia history of the problem records that Abhyankar and Moh used the term for the two-dimensional conjecture only.
Two things keep the claim short of established. First, the printed record does not show Abhyankar and Moh introducing the term: the earliest known published use, in a 1973 paper of Masayoshi Miyanishi, already applies it to the n-dimensional statement, and Bass, Connell and Wright's 1982 survey uses it for arbitrary dimension, so the term was in print for the general statement by the early 1980s, before or alongside Abhyankar's and Moh's own publications. The claim can therefore only be true of a private or oral coinage, which rests on Moh's recollection. Second, the exclusivity ("only") has not been checked against Abhyankar's 1972 lecture notes "Some remarks on the Jacobian question", the one text of his from the period most likely to formulate the problem in several variables, and Moh's 1998 lecture, the very source of the coinage recollection, is reported to define the conjecture for general n. The claim as dated to the 1970s and 1980s survives that later usage, but it shows that Moh at least did not treat the name as reserved for the plane.
What would settle the matter is a reading of Moh's 1998 lecture and of Abhyankar's 1972 and 1977 lecture notes: a passage in either applying the name to the general statement would contradict the "only", while confirmation that Moh's recollection concerns the planar problem would move the claim toward verified.
Full reasoning: the evidence and decisions behind this verdict
This pass re-examined the claim five days after the previous assessment. The evidence landscape moved modestly, mostly through corroboration of the coinage anecdote and sharper dating of the competing usage.
Direct assertions. The claim is asserted outright only by an anonymous commenter (30 July 2026) on Tao's post digesting the counterexample (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/), with no evidence offered. The same thread, read whole this pass, contains three further relevant voices: a second anonymous comment (2 August) giving the anecdote that Abhyankar and Moh began using the name privately for the two-dimensional conjecture after Zariski told them in the 1960s that it was unnamed; a third (31 July) stating that the first published use of "Jacobian conjecture" was Miyanishi's in 1973, for the general case, and that Abhyankar and Moh only began publishing on the planar problem in the later 1970s, while still granting that they made the name refer to the two-variable case; and a fourth noting that Moh's 1998 paper defines the conjecture parametrized by n. Tao's reply argues there was no pre-2026 consensus that the planar case was fundamental, citing Smale's Problem 16, Keller's 1939 paper (which he notes formulates the problem in all dimensions), and van den Essen's 1997 papers; it does not deny what Abhyankar and Moh themselves meant.
Corroboration. The Wikipedia article (revision of 11 September 2026, en.wikipedia.org/wiki/Jacobian_conjecture), read in full, says the coinage is unclear but relays (i) Moh's 1998 lecture (Algebra and Geometry, Taipei 1995, International Press, p. 107): after Zariski's late-1960s Purdue remark, "we decided to call it the Jacobian Conjecture"; (ii) Borisov's 2020 attribution of the coinage to Abhyankar, whose 1977 notes present planar results from 1970–71; (iii) Miyanishi, "Some remarks on polynomial rings", Osaka J. Math. 10 (1973), p. 617, as the earliest printed use, referring to the n-dimensional conjecture; and (iv) that Abhyankar (1977), Moh (1983) and Yitang Zhang (1991) used "Jacobian conjecture" or "Jacobian problem" for the two-dimensional conjecture only. A search snippet of Miyanishi's paper confirms it discusses "the Jacobian conjecture" alongside Zariski's problem and Serre's conjecture, though the dimension could not be checked from the snippet. Moh's 1998 lecture, Miyanishi 1973 and Abhyankar's notes are PDFs that the reading tool could not open in this pass (a known failure on PDF sources), so the Wikipedia footnotes stand in for them; the assessment depends on that secondary relay to that extent.
Weighing. The coinage part, previously resting on one anonymous anecdote, now has Moh's own published recollection behind it (at one remove), which is the best kind of evidence available for a private naming. That strengthens the attribution of the name to Abhyankar as far as a joint Abhyankar–Moh coinage goes. The two-variable focus, Abhyankar's and Moh's planar work in the 1970s and 1980s, is confirmed by the bibliographic record and now by the Wikipedia history explicitly. Against the strong reading, the printed use of the term for the n-dimensional statement now dates to 1973 rather than 1982, which does not falsify a claim about Abhyankar's and Moh's own usage but does mean "introduced the term" cannot be read as a claim about the published record. The report that Moh's 1998 lecture defines the conjecture for general n falls outside the claim's stated period but weakens the picture of Moh as treating the name as exclusively planar.
Unresolved. Abhyankar's 1972 notes "Some remarks on the Jacobian question" (published 1994, Proc. Indian Acad. Sci. 104) were not opened; if they formulate and name the general problem, the "only" fails. Whether Moh's recollection at p. 107 specifies the planar statement was not checked. The Rodríguez Díaz paper "On the origin of the Jacobian conjecture" (arxiv.org/html/2512.23614), flagged last pass as promising, was read and turns out to concern Kraus's 1884 anticipation of the planar statement; it says nothing about the naming and does not bear on this claim.
Verdict unchanged: supported, with slightly higher confidence than before because the coinage anecdote now traces to Moh himself rather than to an anonymous comment. Credence about 0.6: the joint coinage and planar meaning are probable, but the exclusivity rests on unexamined primary texts, one of which (Moh 1998) is reported to use the name for general n. Reading Moh 1998 and Abhyankar 1972 would move the verdict decisively in one direction or the other.
Decomposition
The claims this one rests on directly. ↗︎ opens a subclaim; the map shows how they fit together.
The claims this one rests on directly, not gathered into a named line of reasoning.
- requiresa load-bearing premise: the parent is false without itsteward instructions →Shreeram Abhyankar coined the term "Jacobian conjecture." ↗︎
- supportsthis provides evidence for the parentsteward instructions →Abhyankar and Moh treated the two-variable case of the Jacobian problem in the 1970s and 1980s. ↗︎
- contradictsthis argues against the parentsteward instructions →By the early 1980s the term "Jacobian conjecture" was used in print for the n-dimensional statement. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
The only source that states the claim outright is an anonymous comment on Terence Tao's blog, offered without evidence in a dispute over which version of the conjecture the 2026 counterexample refuted; a second comment in the same thread supplies the anecdote behind it (Abhyankar and Moh adopting the name privately after a remark of Zariski), and a third replies that the earliest printed use of the term, by Miyanishi in 1973, referred to the n-dimensional statement. The corroboration comes secondhand through the Wikipedia history section, which relays Moh's own 1998 recollection ("we decided to call it the Jacobian Conjecture") and the two-variable focus of Abhyankar's 1977 notes and Moh's 1983 paper, while explicitly declining to say who coined the name. The primary documents that would settle the matter, Moh's 1998 lecture and Abhyankar's 1972 and 1977 lecture notes, have not been opened, and a reader should treat the confident wording of the blog assertion as resting on that unexamined testimony.
But the fact is, when Abhyankar and Moh named "Jacobian conjecture" in the 1970-80s, they only referred to the 2-dimensional version.
An extended comment thread debates whether the two-dimensional or the general n-dimensional statement is the true 'Jacobian conjecture'; Tao replies that prior to 2026 there was no consensus that the planar case was the more fundamental one, citing Smale, Keller, van den Essen, and Wikipedia.
Asserted without evidence of the source's own. The assertion is made flatly by an anonymous commenter, with no source offered. Elsewhere in the same thread a second anonymous commenter supplies the supporting anecdote (Abhyankar and Moh adopting the name privately after a remark of Zariski in the 1960s), and a third replies that the earliest printed use of the term, by Miyanishi in 1973, referred to the n-dimensional statement and predates Abhyankar's and Moh's publications on the planar problem. Tao's reply disputes the surrounding contention that the planar case was consensually the fundamental one, without addressing what Abhyankar and Moh themselves meant. Worth reading closely: The thread holds both the fullest statement of the coinage anecdote and the strongest counterpoint (Miyanishi 1973), plus the observation that Moh's 1998 lecture defines the conjecture for general n; a reader weighing the claim should see all three together.
later recalled that, after Oscar Zariski pointed out at a seminar at Purdue University in the late 1960s that the assertion remained unproved, "we decided to call it the Jacobian Conjecture".
History section of the Wikipedia article (revision of 11 September 2026). The article says it is unclear who coined the name and that the earliest printed use (Miyanishi 1973) referred to the n-dimensional statement, but it relays Moh's recollection of adopting the name with Abhyankar, Borisov's attribution of the coinage to Abhyankar, and separately states that Abhyankar and Moh used the term to refer only to the two-dimensional conjecture. It affirms the components of the claim while stopping short of asserting that Abhyankar and Moh introduced the term.
The source's own evidence bears what it asserts. The encyclopedia article is careful: it says the coinage is not clearly documented, gives Miyanishi's 1973 paper as the earliest printed use (for the n-dimensional statement), and then relays Moh's own recollection of adopting the name after Zariski's remark, Borisov's attribution to Abhyankar, and the fact that Abhyankar's 1977 notes and Moh's 1983 paper apply the name to the two-variable problem. Each statement is footnoted to a checkable source, so the article bears what it says, but it deliberately does not assert that Abhyankar and Moh introduced the term. Worth reading closely: Its footnotes locate the primary sources a decisive pass would open: Moh's 1998 lecture (p. 107) for the coinage recollection, Miyanishi 1973 (p. 617) for the earliest printed use, and Abhyankar's 1977 notes (p. v) for his own framing. The quoted passage was not found in the stored copy of this source.
How these sources relate
- Jacobian conjecture draws its statement from https://www.math.purdue.edu/~ttm/jacobian.pdf. The article's account of who adopted the name rests on Moh's own published recollection. The target is a PDF that could not be opened in this pass, so whether Moh's text specifies that the name was adopted for the two-variable statement, and whether "we" names Abhyankar, could not be checked; the same lecture is reported elsewhere to define the conjecture for general n. Judged from the citing document alone.
Assessment history
0 status changes over 2 assessments. full history →
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.