--- title: Jacobian Conjecture Counterexample | daily.dev description: A shared ChatGPT conversation presents a claimed counterexample to the Jacobian Conjecture. It constructs an explicit polynomial map G from C³ to C³ with... canonical: https://daily.dev/posts/jacobian-conjecture-counterexample-m5wjc367e twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: Jacobian Conjecture Counterexample | daily.dev og:description: A shared ChatGPT conversation presents a claimed counterexample to the Jacobian Conjecture. It constructs an explicit polynomial map G from C³ to C³ with... og:url: https://daily.dev/posts/jacobian-conjecture-counterexample-m5wjc367e og:image: https://api.daily.dev/og/posts/m5WJc367E.png og:image:alt: Jacobian Conjecture Counterexample og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # Jacobian Conjecture Counterexample **[Hacker News](https://daily.dev/sources/hn)** · 2 min read · 0 upvotes · 0 comments ## Summary A shared ChatGPT conversation presents a claimed counterexample to the Jacobian Conjecture. It constructs an explicit polynomial map G from C³ to C³ with constant nonzero Jacobian determinant (det DG = −1), derived by composing a polynomial isomorphism between an affine variety X and A³ with a specific polynomial map F. The conversation verifies the Jacobian is identically −1 via symbolic differentiation and shows the resulting map matches a previously known candidate counterexample up to linear coordinate changes and output rescalings. ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Community take How the wider developer community reacted, aggregated from 2 discussions and 1027 comments across hackernews (as of 2026-07-23). **TL;DR:** The community broadly agrees that advanced mathematics notation and terminology is uniquely dense and hard to penetrate for outsiders, sparking a lively debate over whether this is a notation/communication problem or simply reflects the genuine depth and abstraction of the subject. Many commenters share personal anecdotes of feeling lost, while mathematicians and CS folks push back on oversimplified comparisons to computing complexity. **Sentiment:** 15% positive · 45% mixed · 40% skeptical **The case for** - Several commenters note that compact math notation is optimized for reading and pattern recognition, not just writing, making structural manipulation easier for experts. - A practicing mathematician (daynthelife) concedes that better communication is possible and that many concepts are not as hard as they appear when explained accessibly. - Commenters highlight that tools like Lean and LLMs (AI) are genuinely helping bridge the gap between informal math intuition and formal proof, reducing friction for newcomers. - The concept of 'mathematical maturity' is raised as a real phenomenon — fluency does come with time and exposure, as confirmed by those who studied math. **The pushback** - Many commenters argue that math's impenetrability is not just notation but reflects a massive prerequisite DAG — you need years of foundational study before advanced concepts become comprehensible. - Several note that even professional mathematicians feel lost outside their own narrow specialty, suggesting the density is intrinsic, not just a communication failure. - A pure math PhD (jebarker) warns that advanced math knowledge is catastrophically forgettable — even their own thesis is impenetrable to them 20 years later. - Some push back on the idea that CS terminology is comparably dense, arguing math's abstraction is fundamentally more removed from lived experience than computing concepts. - The claim that math 'strives to minimize ambiguity' is contested — terms like 'normal' have ~20 different meanings and notation is largely passed down without standardization efforts. **By community** - hackernews (mixed): Commenters are split between sympathy for math's impenetrability (seen as a real barrier) and pushback that it reflects genuine depth rather than poor communication, with lively sub-debates on notation, CS vs. math difficulty, and the role of LLMs/Lean. **Hottest debate:** Whether advanced mathematics is hard primarily because of poor notation/communication or because of its genuinely deep and abstract conceptual content — with CS vs. math difficulty comparisons generating the most back-and-forth. **Open questions** - Could better pedagogical tools (LLMs, proof assistants like Lean, code-based explanations) meaningfully lower the barrier to advanced mathematics for non-specialists? - Is there a realistic path to standardizing or improving mathematical notation without sacrificing the expressive power experts rely on? - How much of the perceived gap between CS and math difficulty is due to accumulated tacit knowledge that CS practitioners don't recognize they have? **Highlights** > This is perhaps more a comment on Wikipedia's coverage of mathematics. They have some general guidelines in their manual of style, but it's really hard to write math articles in a way such that something like the Rees algebra without defining 1000 thinks beforehand. To understand the definition of the Rees algebra, you would need to define, mostly in order: sets, groups, abelian groups, rings, ideals of rings, algebras over rings, direct sums of rings, adjoining things to rings, etc. This is just to understand the definition; to understand its significance in algebraic geometry (which I have no idea of), there are a thousand more definitions. The issue with trying to understand a concept in math is there is a massive directed acyclic graph of prerequisites leading to these concepts, and one needs to traverse this graph in the right order. Unfortunately, knowing the right order is almost tantamount to understanding the concept itself. > — [physidev on hackernews](https://news.ycombinator.com/item?id=49015267) > As mathematician, I actually agree. We could do a lot more to communicate ideas in a way non-experts can understand. For instance, most people are familiar with polynomials. Take a polynomial in x (with integer coefficients) and substitute x for 5t everywhere. So for instance, 2x^3 - 8x + 3 becomes 250t^3 - 40t + 3. The Rees algebra Z[5t] (here Z is the integers) is then just the set of all polynomials you can get this way. If Wikipedia introduced it like this, I don't think most people would have a problem understanding. The concept is not (at least not always) actually that complicated, like others are implying. It's our communication that is lacking. > — [daynthelife on hackernews · 2 comments](https://news.ycombinator.com/item?id=49018123) > I second this and would add that it's really easy to catastrophically forget things in math. I'm pretty sure that most CS knowledge I have I will retain at a level where I won't forget the general ideas and re-reading materials can quickly refresh the details. This is not true for advanced math. I did a pure math PhD and my own thesis is impenetrable to me 20 years later. It would take months, if not years, of focused effort for me to regain the understanding. > — [jebarker on hackernews](https://news.ycombinator.com/item?id=49014373) > I think you are snagging on thinking this is an observation about difficulty, time-to-mastery, or mental firepower requirements. It's not. It's a plain observation that math exists on mostly it's own path with little to zero overlap with our lived experiences. If mathematics was a vector, it would have similar magnitude to other vectors, but it's direction would be much more removed from the typical knowledge pack, forcing you to get really close to the origin before you can "hop" over to that math vector. Other "knowledge" vectors, by virtue of being more bunched up, are closer together much further up, if that poor analogy at all makes sense. > — [WarmWash on hackernews · 4 comments](https://news.ycombinator.com/item?id=49013971) > I'm sure you're right that there's poor terminology, and Wikipedia math is certainly daunting. But in general the many mathematical exposition PDFs scattered across the Internet are beautifully written by careful, intelligent people. I'm a software engineer who's spent many hours reading them and also trying to do graduate classes as an adult. It's unfortunately common on Hacker News to encounter people who think that "I'd be good at math if only they used code to explain it" or "I'd be good at math if it used better terminology/notation". The truth is that math is hard: few other subjects have anything close to its crazy conceptual breadth and depth, with hardish concepts being built upon hardish concepts in many layers. > — [Myrmornis on hackernews](https://news.ycombinator.com/item?id=49020543) **Source threads** - [hackernews](https://news.ycombinator.com/item?id=49010345) · 815 points · 1027 comments - [hackernews](https://news.ycombinator.com/item?id=49002013) · 7 points · 0 comments ## Similar posts on daily.dev - [Don’t just attend KubeCon \+ CloudNativeCon, Merge Forward your experience\!](https://daily.dev/posts/don-t-just-attend-kubecon-cloudnativecon-merge-forward-your-experience--l0rpp73x8) · CNCF · 1 upvotes · 0 comments - [Announcing H2 2026 KCDs](https://daily.dev/posts/announcing-h2-2026-kcds-m96goajm1) · CNCF · 1 upvotes · 0 comments - [Two months of Open Community Groups](https://daily.dev/posts/two-months-of-open-community-groups-asf52zhbs) · CNCF · 0 upvotes · 0 comments - [CNCF Unveils Schedule for KubeCon \+ CloudNativeCon Europe 2026](https://daily.dev/posts/cncf-unveils-schedule-for-kubecon-cloudnativecon-europe-2026-ikhcoa5cb) · CNCF · 2 upvotes · 0 comments - [CNCF Debuts KubeCon \+ CloudNativeCon Japan 2026 Schedule](https://daily.dev/posts/cncf-debuts-kubecon-cloudnativecon-japan-2026-schedule-xp5pyudub) · CNCF · 1 upvotes · 0 comments --- Tags: [#chatgpt](https://daily.dev/tags/chatgpt), [#math](https://daily.dev/tags/math) [View this post on daily.dev](https://daily.dev/posts/jacobian-conjecture-counterexample-m5wjc367e) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"Jacobian Conjecture Counterexample","url":"https://daily.dev/posts/jacobian-conjecture-counterexample-m5wjc367e","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/jacobian-conjecture-counterexample-m5wjc367e"},"datePublished":"2026-07-22T18:12:49.363Z","dateModified":"2026-07-23T12:39:19.916Z","description":"A shared ChatGPT conversation presents a claimed counterexample to the Jacobian Conjecture. 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