Unsolved Problem by Fields Medalist Breached by Two High School Students

marsbitPublished on 2026-09-14Last updated on 2026-09-14

Abstract

In a remarkable breakthrough, two high school students from Oak Park High School, Aayush Bathija and Prince Rohatgi, with guidance from UCLA postdoctoral researcher Daniel Soskin, have solved an open problem related to the theory of Lorentzian polynomials—a field pioneered by Fields Medalist June Huh. Their 75-page paper, "Bounded Ratios for Lorentzian Polynomials," is now available on arXiv. The research investigates the constraints on coefficient ratios within Lorentzian polynomials, extending prior work that characterized bounded ratios for quadratic polynomials to polynomials of arbitrary degree. The core challenge was determining which coefficient ratios have a universal upper bound across all polynomials in the class and identifying the optimal bounds. The students heavily utilized AI assistants, specifically Claude Opus 5 and GPT-5.6 Sol, for computational exploration, proof idea generation, and editing. They emphasize that while AI provided helpful suggestions, some were misleading, and all calculations and arguments were independently verified. This achievement is particularly notable as it follows a recent open letter signed by 25 Fields Medalists, including June Huh, expressing concerns about AI's impact on mathematical rigor. The students' work demonstrates how AI can lower barriers to entry, enabling talented pre-university researchers to engage with advanced mathematical frontiers.

• #AI
• #arXiv
• #Collaboration
• #Fields Medal
• #High School
• #Lorentzian
• #Mathematics
• #Polynomials
• #Proof
• #Research

Can you believe it?

They haven't even started university, yet their research has connected with the work of a "Fields Medalist".

Just today, stunning news spread through the UCLA mathematics community—

Two 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.

Even more eye-catching is that this proof was completed with heavy assistance from AI!

One was Claude Opus 5, the other GPT-5.6 Sol.

Currently, the paper titled "Bounded Ratios for Lorentzian Polynomials" has been published on arXiv, spanning 75 pages.

Paper: https://arxiv.org/pdf/2609.05341

The names of the two high school students and one postdoc are written right there in the author list.

The high school students' mentor, UCLA Postdoctoral Fellow Daniel Soskin

So, what problem exactly did they solve in this research?

To clarify this, we must first start with the work of Fields Medalist June Huh.

What Did They Actually Solve?

June Huh is Korean and won the Fields Medal in 2022.

His 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).

In 2020, the theory of "Lorentzian polynomials" that he and his collaborators established was one of his important representative works.

The name might sound unfamiliar, but "polynomials" are something everyone encountered in middle school. The numbers in front of each term are the coefficients.

The special feature of "Lorentzian polynomials" is that there are strict mathematical constraints between these coefficients; they cannot be paired arbitrarily.

This theory connects problems in combinatorics with geometry and inequalities between coefficients.

This research inquires about precisely how strong these constraints are.

For example, if you multiply some coefficients and divide by others, you get a ratio.

As long as the polynomial satisfies the specified conditions, does this ratio have an "impenetrable ceiling"?

Do you still remember the definition of the classic Hessian matrix from advanced calculus?

Calculating 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.

If there is an upper bound, we must further ask: How low can this upper bound be pushed to obtain the most precise answer?

Previously, 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.

But this path was not yet complete.

Moving from quadratic to cubic, quartic, and even arbitrary degrees, do the old rules still hold?

The higher the degree, the more complex the relationships between coefficients become. Conclusions that hold in the quadratic case cannot be directly transferred.

This is the gap where the two high school students entered—

advancing the research on coefficient ratios to higher degrees, clarifying which ratios always have bounds, and further seeking the most precise upper limits.

The 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.

Reading this far, you just need to remember one question: How high can these constrained coefficients push the ratio?

Next, we'll use a simple example to explain this "ceiling" clearly.

Lorentzian Polynomials: "Balance Constraints" Between Coefficients

First, look at a polynomial:

Here a, b, c are all positive numbers. Writing the middle term as 2b is to make the subsequent relationship cleaner.

In this simplest case, it is a Lorentzian polynomial precisely requiring:

In 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.

For 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.

When 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.

Transforming the previous inequality slightly:

This is called a "bounded ratio": no matter how you choose coefficients that meet the conditions, this ratio cannot exceed 1.

But in reverse:

there is no uniform upper bound.

For instance, fix a=c=1, let b continuously increase, still satisfying the Lorentzian condition, but the inverse ratio can become arbitrarily large.

Naturally, one would ask:

Facing more complex polynomials, when multiplying and dividing different coefficients, which combinations have an upper bound, and which do not?

This paper aims to systematically answer this question, with the previous quadratic example belonging to the existing foundational case.

Two High School Students, Teamed Up with AI, Cracked It

The two students who wrote this paper, Aayush Bathija and Prince Rohatgi, both attend Oak Park High School in California.

They are also members of the UCLA Olga Radko Math Circle (ORMC).

Aayush 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.

In 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.

But this time, under the guidance of UCLA mathematics postdoctoral fellow Daniel Soskin, they directly teamed up with AI to wage a brilliant blitzkrieg.

https://circles.math.ucla.edu/circles/index.shtml

The core of their problem-solving was deeply embedding AI and formal computational tools into the central workflow of mathematical discovery.

The 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.

Transforming Complex Numerical Values into Simple Exponents

Let t approach 0, with the three coefficients taking values:

Then:

In reverse, it's t^-4, which tends towards infinity.

The 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.

This 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.

You can think of it as a mathematical rule for "which rates of change can appear simultaneously."

A 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.

Therefore, checking these paths can be upgraded into a complete criterion. The proof uses tools from semi-algebraic geometry like curve selection lemmas.

And don't forget, all this sophisticated talk was accomplished by two high school students with the help of AI.

Claude and ChatGPT, Participated in "Proof Idea Generation"

In the paper's acknowledgments, the authors lay their cards on the table—

Claude Opus 5 and GPT-5.6 Sol were the core tools for this proof.

They were used for computation, proof idea generation, and editing assistance. Among these, "proof idea generation" is especially crucial.

This means the models' use has deepened into research exploration: facing an unsolved problem, searching for possible derivation paths that might hold.

However, the authors also clearly wrote the other side: Some suggestions were helpful, others were misleading.

They stated they have independently verified all calculations and take full responsibility for the paper's content.

These 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.

When 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.

AI has entered the exploration phase, but rigorous verification remains the prerequisite for valid results.

The 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.

Fields Medalists Collectively Call for a Halt, But It Can't Be Stopped

Dramatically, just one day before this research was announced, the mathematics world was rocked by a major uproar.

Twenty-five Fields Medalists from around the world jointly signed an open letter, issuing the strongest warning: "AI is destroying mathematics itself!"

June Huh himself was one of the signatories.

Partial list

Who would have thought that two high school students, using AI assistance, produced results in June Huh's own research direction.

The 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.

But the story of Aayush and Prince presents another side—

After 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.

An AIME-qualified high school student, guided by a postdoc, using AI assistance, touched a path a Fields Medalist hadn't finished walking.

Five years ago, this was almost impossible.

The frontiers of mathematics no longer belong only to those who have spent over a decade just to reach the doorstep.

Whether this is good or bad, the mathematical community will debate for a long time. But the door has already been pushed open a crack.

References:

https://circles.math.ucla.edu/circles/index.shtml

https://arxiv.org/pdf/2609.05341

This article is from WeChat Official Account "新智元" (ID: AI_era), authors: Taozi, Mark