{"slug": "unsolved-problem-by-fields-medalist-breached-by-two-high-school-students", "title": "Unsolved Problem by Fields Medalist Breached by Two High School Students", "summary": "Two high school students, Aayush Bathija and Prince Rohatgi, working with UCLA postdoctoral fellow Daniel Soskin, published a 75-page arXiv paper titled \"Bounded Ratios for Lorentzian Polynomials\" that extends coefficient-ratio bounds for Lorentzian polynomials from the quadratic case to arbitrary degrees, a problem Fields Medalist June Huh had not solved. The proof was completed with heavy assistance from AI models Claude Opus 5 and GPT-5.6 Sol, according to the report. The paper's Main Structural Theorem shows whether a coefficient ratio has a uniform upper bound can be determined by a set of discrete convexity conditions.", "body_md": "Can you believe it?\n\nThey haven't even started university, yet their research has connected with the work of a \"Fields Medalist\".\n\nJust today, stunning news spread through the UCLA mathematics community—\n\nTwo high school students, Aayush Bathija and Prince Rohatgi, under the guidance of a postdoctoral researcher, have solved a problem that mathematician June Huh never managed to crack.\n\nEven more eye-catching is that this proof was completed with heavy assistance from AI!\n\nOne was Claude Opus 5, the other GPT-5.6 Sol.\n\nCurrently, the paper titled \"Bounded Ratios for Lorentzian Polynomials\" has been published on arXiv, spanning 75 pages.\n\nPaper: https://arxiv.org/pdf/2609.05341\n\nThe names of the two high school students and one postdoc are written right there in the author list.\n\nThe high school students' mentor, UCLA Postdoctoral Fellow Daniel Soskin\n\nSo, what problem exactly did they solve in this research?\n\nTo clarify this, we must first start with the work of Fields Medalist June Huh.\n\n## **What Did They Actually Solve?**\n\nJune Huh is Korean and won the Fields Medal in 2022.\n\nHis journey is quite legendary: he dropped out of high school to become a poet before switching to pure mathematical research, winning the Fields Medal at the age of 39 (the Fields Medal is only awarded to individuals under 40).\n\nIn 2020, the theory of \"Lorentzian polynomials\" that he and his collaborators established was one of his important representative works.\n\nThe name might sound unfamiliar, but \"polynomials\" are something everyone encountered in middle school. The numbers in front of each term are the coefficients.\n\nThe special feature of \"Lorentzian polynomials\" is that **there are strict mathematical constraints between these coefficients; they cannot be paired arbitrarily.**\n\nThis theory connects problems in combinatorics with geometry and inequalities between coefficients.\n\nThis research inquires about precisely how strong these constraints are.\n\nFor example, if you multiply some coefficients and divide by others, you get a ratio.\n\n**As long as the polynomial satisfies the specified conditions, does this ratio have an \"impenetrable ceiling\"?**\n\nDo you still remember the definition of the classic Hessian matrix from advanced calculus?\n\nCalculating the ratio for a specific set of coefficients is not difficult. The real challenge is: facing an infinite variety of polynomials that meet the conditions, determining which ratios always have an upper bound and which can increase indefinitely.\n\nIf there is an upper bound, we must further ask: **How low can this upper bound be pushed to obtain the most precise answer?**\n\nPreviously, June Huh and his collaborators had characterized which ratios have upper bounds for **quadratic Lorentzian polynomials** and had found the optimal upper bounds for the case of three variables.\n\nBut this path was not yet complete.\n\n**Moving from quadratic to cubic, quartic, and even arbitrary degrees, do the old rules still hold?**\n\nThe higher the degree, the more complex the relationships between coefficients become. Conclusions that hold in the quadratic case cannot be directly transferred.\n\nThis is the gap where the two high school students entered—\n\n**advancing the research on coefficient ratios to higher degrees, clarifying which ratios always have bounds, and further seeking the most precise upper limits.**\n\nThe paper's core formula, the \"Main Structural Theorem,\" extends the relevant conclusions from previous quadratic Lorentzian polynomials to arbitrary degrees. It shows that whether a coefficient ratio has a uniform upper bound can be completely determined using a set of discrete convexity conditions.\n\nReading this far, you just need to remember one question: **How high can these constrained coefficients push the ratio?**\n\nNext, we'll use a simple example to explain this \"ceiling\" clearly.\n\n### **Lorentzian Polynomials: \"Balance Constraints\" Between Coefficients**\n\nFirst, look at a polynomial:\n\nHere a, b, c are all positive numbers. Writing the middle term as 2b is to make the subsequent relationship cleaner.\n\n**In this simplest case, it is a Lorentzian polynomial precisely requiring:**\n\nIn other words: **If the coefficients at both ends are large, the coefficient in the middle cannot be too small.** This is an introductory example of the log-concave property embodied by the Lorentzian structure.\n\nFor example, if a=4, c=9, then b must be >=6. You cannot arbitrarily change b to 1 and still expect it to remain in this polynomial family.\n\nWhen there are more variables and higher degrees, these constraints become more complex, requiring various derivative results to collectively satisfy specific conditions. The formal definition involves a matrix describing curvature, which in the strict case has an eigenvalue structure with \"one positive, the rest negative.\" **The name \"Lorentzian\" corresponds to this positive-negative structure**. However, simply remembering that \"the coefficients are subject to a complete set of interrelated constraints\" is enough to understand the main thread of this article.\n\nTransforming the previous inequality slightly:\n\nThis is called a **\"bounded ratio\"**: no matter how you choose coefficients that meet the conditions, this ratio cannot exceed 1.\n\nBut in reverse:\n\nthere is no uniform upper bound.\n\nFor instance, fix a=c=1, let b continuously increase, still satisfying the Lorentzian condition, but the inverse ratio can become arbitrarily large.\n\nNaturally, one would ask:\n\n**Facing more complex polynomials, when multiplying and dividing different coefficients, which combinations have an upper bound, and which do not?**\n\nThis paper aims to systematically answer this question, with the previous quadratic example belonging to the existing foundational case.\n\n## **Two High School Students, Teamed Up with AI, Cracked It**\n\nThe two students who wrote this paper, Aayush Bathija and Prince Rohatgi, both attend Oak Park High School in California.\n\nThey are also members of the UCLA Olga Radko Math Circle (ORMC).\n\nAayush is a sophomore and an AIME qualifier. Prince is currently a junior, also having made AIME, and serves as a main instructor for AMC 10/12 (equivalent to high school freshman) competition classes within the math circle.\n\nIn the past, for a high school student to even touch a \"Fields Medal-level\" field, they would need at least 5-8 years of university and doctoral training.\n\nBut this time, under the guidance of UCLA mathematics postdoctoral fellow Daniel Soskin, they directly teamed up with AI to wage a brilliant blitzkrieg.\n\nhttps://circles.math.ucla.edu/circles/index.shtml\n\nThe core of their problem-solving was deeply embedding AI and formal computational tools into the central workflow of mathematical discovery.\n\nThe overall core idea of the proof can be understood as: **To check whether a ratio might run out of control, first study how it changes under extreme conditions.**\n\n### **Transforming Complex Numerical Values into Simple Exponents**\n\nLet t approach 0, with the three coefficients taking values:\n\nThen:\n\nIn reverse, it's t^-4, which tends towards infinity.\n\nThe most useful point here is: **Multiplication and division become addition and subtraction of exponents. The sign of the exponent directly tells us whether the ratio will shrink or explode.**\n\nThis idea of extracting the \"dominant power\" is related to the **\"tropicalization\" method** used in the paper. Existing Lorentzian polynomial theory tells us: legitimate power patterns are governed by a discrete convexity rule called **M-convexity**.\n\nYou can think of it as a mathematical rule for \"which rates of change can appear simultaneously.\"\n\nA key step in the paper is to prove: **If a ratio can increase indefinitely, a path with this power characteristic can be found to expose the runaway.**\n\nTherefore, checking these paths can be upgraded into a complete criterion. The proof uses tools from semi-algebraic geometry like curve selection lemmas.\n\nAnd don't forget, all this sophisticated talk was accomplished by two high school students with the help of AI.\n\n### **Claude and ChatGPT, Participated in \"Proof Idea Generation\"**\n\nIn the paper's acknowledgments, the authors lay their cards on the table—\n\nClaude Opus 5 and GPT-5.6 Sol were the core tools for this proof.\n\nThey were used for **computation, proof idea generation, and editing assistance**. Among these, \"proof idea generation\" is especially crucial.\n\nThis means the models' use has deepened into research exploration: facing an unsolved problem, searching for possible derivation paths that might hold.\n\nHowever, the authors also clearly wrote the other side: **Some suggestions were helpful, others were misleading.**\n\nThey stated they have independently verified all calculations and take full responsibility for the paper's content.\n\nThese few sentences outline the actual boundaries of this collaboration: AI can provide ideas, researchers need to screen, scrutinize, and turn valuable clues into arguments that can withstand scrutiny.\n\nWhen the model gives a seemingly beautiful answer, the research is not over. Whether conditions are complete, derivations are valid, edge cases are missed—all must be further questioned.\n\n**AI has entered the exploration phase, but rigorous verification remains the prerequisite for valid results.**\n\nThe mentorship of the postdoctoral advisor and the support of the ORMC research program are thus particularly important. The two students were able to access genuine open problems and conduct research under professional guidance. AI was embedded into this collaboration, assisting with computation and idea exploration.\n\n## **Fields Medalists Collectively Call for a Halt, But It Can't Be Stopped**\n\nDramatically, just one day before this research was announced, the mathematics world was rocked by a major uproar.\n\nTwenty-five Fields Medalists from around the world jointly signed an open letter, issuing the strongest warning: \"AI is destroying mathematics itself!\"\n\n**June Huh himself was one of the signatories.**\n\nPartial list\n\nWho would have thought that two high school students, using AI assistance, produced results in June Huh's own research direction.\n\nThe 25 Fields Medalists' concern is that the rigor of pure mathematical research and the purity of scientific inquiry might be crushed by computational power and algorithms.\n\nBut the story of Aayush and Prince presents another side—\n\nAfter AI lowers the research threshold, what's being unleashed is not just \"problem-solving speed,\" but also more people's eligibility to enter the field.\n\nAn AIME-qualified high school student, guided by a postdoc, using AI assistance, touched a path a Fields Medalist hadn't finished walking.\n\nFive years ago, this was almost impossible.\n\nThe frontiers of mathematics no longer belong only to those who have spent over a decade just to reach the doorstep.\n\nWhether this is good or bad, the mathematical community will debate for a long time. But the door has already been pushed open a crack.\n\n### References:\n\nhttps://circles.math.ucla.edu/circles/index.shtml\n\nhttps://arxiv.org/pdf/2609.05341\n\nThis article is from WeChat Official Account \"新智元\" (ID: AI_era), authors: Taozi, Mark", "url": "https://wpnews.pro/news/unsolved-problem-by-fields-medalist-breached-by-two-high-school-students", "canonical_source": "https://www.htx.com/en-in/news/internet-shocked-unsolved-problem-by-fields-medalist-breache-IBDgZG2j/", "published_at": "2026-09-14 12:23:11+00:00", "updated_at": "2026-09-14 12:39:31.229193+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research", "large-language-models", "ai-tools"], "entities": ["Aayush Bathija", "Prince Rohatgi", "Daniel Soskin", "June Huh", "UCLA", "Claude Opus 5", "GPT-5.6 Sol", "arXiv"], "alternates": {"html": "https://wpnews.pro/news/unsolved-problem-by-fields-medalist-breached-by-two-high-school-students", "markdown": "https://wpnews.pro/news/unsolved-problem-by-fields-medalist-breached-by-two-high-school-students.md", "text": "https://wpnews.pro/news/unsolved-problem-by-fields-medalist-breached-by-two-high-school-students.txt", "jsonld": "https://wpnews.pro/news/unsolved-problem-by-fields-medalist-breached-by-two-high-school-students.jsonld"}}