arXiv subjects

Ruyong Feng

Publications and source records attributed to Ruyong Feng.

At least 19 recordsLinked to original sources

arXiv2026

MechMath Agent Team: LLM Driven Agents for Mathematical Research

AI reasoning has become a central focus in contemporary artificial intelligence, largely driven by the success of large language models. However, mathematical research, which is characterized by non-linear derivation paths, rigorous logical requirements, and protracted exploration cycles, poses severe challenges for existing reasoning systems. To overcome these limitations, we present the MechMath Agent Team (MMAT), which is a large language model driven agent designed to serve as a co-pilot throughout the full cycle of mathematical research. We design a tripartite Harness Architecture that decouples system responsibilities into Control, Execution, and Augmentation planes, thereby reconciling rigorous logical control with the agility demanded by open-ended research. Building upon this framework, we instantiate three specialized agents: a Knowledge Base Manager, a Natural Language Prover, and a Formal Language Prover, all operating in a closed loop to produce formally certified mathematical proofs. We evaluate MMAT on open problems in Number Theory, Algebraic Complexity Theory, Differential Algebra, Operator Algebra, and Inequalities. Across a two-month deployment, 11 problems have been solved, demonstrating its capacity to act as a co-pilot throughout the entire research cycle. The contributions are threefold: a general decoupled Harness Architecture for multi-agent mathematical reasoning, its concrete instantiation in the MMAT system, and empirical validation on a diverse suite of open problems.

cs.AI↗

arXiv2026

The Equivalence Problem for Generalized Airy Operators

In this paper, we establish degree obstructions to the equivalence of generalized Airy operators of the same type. As an application, we answer a question posed by Nicholas M. Katz in Inventiones Mathematicae (87, pp. 13-61,1987). The main results of Sections 3 and 4 were obtained through a close interactive collaboration between the authors and the artificial intelligence agent system MechMath Agent Team (MMAT).

math.RA↗

arXiv2026

Every Nonnegative Integer Is a Sum of a Triangular, a Pentagonal, and a Heptagonal Number

In this paper, it is proved that any nonnegative integer can be written in the following form $$ x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2, \qquad x,y,z \in \mathbb{N}. $$ This settles the conjecture recorded as OEIS A287616. All parts of the proof have been formalized in Lean 4, with the exception of two results: one externally cited theorem and one statement verified by symbolic computation. Both the natural-language proof and the Lean formalization were generated by the MechMath Agent Team developed by the authors.

math.NT↗

arXiv2026

A Greatest Common Divisor Criterion of Certain Binomial Coefficients

The binomial greatest common divisor (gcd) criterion recorded as OEIS A080170 is proven. The criterion also appears as conjecture (17) in Ralf Stephan's list of OEIS conjectures. For $k\geq 2$, put \[ D(k)=\gcd_{2\leq q\leq k+1}\binom{qk}{k}, \qquad n=k+1. \] If $P$ is the largest prime-power component $p^a$ exactly dividing $n$, then the criterion asserts \[ D(k)=1 \quad\Longleftrightarrow\quad \frac{n}{P}>P. \] The proof is formalized in Lean and the Lean artifact is accepted as part of the Formal Conjectures project. Both the natural-language proof and the Lean formalization are generated by the MechMath Agent Team, an AI agent developed by the authors.

math.NT↗

arXiv2026

Sparse Polynomial Divisibility Test over Finite Field is CoNP-hard

In this paper, we show that deciding whether a sparse polynomial does not divide another sparse polynomial exactly over finite fields is NP-hard under BPP many-one reductions. Equivalently, the sparse polynomial divisibility test over finite fields is CoNP-hard. This resolves the long-standing open problem concerning the computational complexity of the divisibility test for sparse polynomials in the setting of finite fields.

cs.SC↗

arXiv2026

Output-sensitive Sparse Polynomial GCD over Finite Fields is NP-hard

In this paper, we prove that output-sensitive sparse polynomial GCD computation over finite fields is NP-hard under BPP many-one reduction. More precisely, for two sparse univariate polynomials $f,g$ with finite field coefficients, there exists no randomized algorithm to compute $\mathrm{gcd}(f,g)$, which is polynomial-time in the sizes of $f,g,\gcd(f,g)$ under the standard complexity assumption $\mathrm{NP}\nsubseteq\mathrm{BPP}$. This settles the open problem posed as Challenge 5 in The Sparsity Challenges in the finite field setting. Furthermore, we show that the Roots of Unity Detection problem over finite fields is NP-hard; that is, determining whether the GCD of a sparse univariate polynomial and $x^n - 1$ has nonzero degree is NP-hard.

cs.SC↗

arXiv2026

A Finite Certificate for the Positive $n=9$ Vasc Inequality

We prove the positive-real $n=9$ case of the Vasc cyclic inequality. The proof was obtained with human-guided assistance from the AI agent MechMath Agent Team: the human-readable part reduces the rational inequality to a homogeneous polynomial inequality, fixes a cyclic maximum, and parametrizes each sorted fixed-maximum cone by cumulative gaps; the finite part is a certificate covering all $8!=40320$ sorted cones. MechMath Agent Team generated the certificate verification workflow through Python tool calls, including the case split, verification programs, and terminal classifications. The published certificate has $36815$ coefficient leaves, $2236$ ordinary Polya multiplier leaves, and $1269$ AM-GM midpoint overlay leaves. Human authors audited the mathematical reductions and verification logic, and a separate artifact contains the certificate, an independent verifier, and a from-source rebuild route.

cs.SC↗

arXiv2026

MechMath: Sorrifier-Driven Formal Decomposition Workflow for Automated Theorem Proving

Recent advances in large language models (LLMs) and LLM-based agents have substantially improved the capabilities of automated theorem proving. However, for problems that require complex mathematical reasoning, current systems seldom succeed in their initial attempt, necessitating iterative adjustments to their proof strategies. Existing approaches for handling failed attempts typically either iteratively fix errors within the proof or discard the entire proof and regenerate it from scratch. The former leads to progressively longer contexts, which degrade the model's ability to attend to the remaining unresolved subproblems, while the latter is inefficient, as it may abandon mostly correct reasoning due to localized errors. To address this dilemma, we present MechMath, an agent system centered on a Sorrifier-driven formal decomposition paradigm. By leveraging the sorry placeholder in Lean to precisely isolate unresolved subgoals while preserving the surrounding verified proof structure, MechMath extracts each failed subproblem into a clean, self-contained context and resolves it independently. This avoids both the waste of full regeneration and the excessive context length induced by repeated repairs. Experimental results on challenging mathematical competition benchmarks, including IMO 2025, Putnam 2025, miniF2F, and a subset of ProverBench, demonstrate that our agent achieves significant advantages in proving efficiency.

cs.CL↗

arXiv2024

Parallel Summation in P-Recursive Extensions

We propose investigating a summation analog of the paradigm for parallel integration. We make some first steps towards an indefinite summation method applicable to summands that rationally depend on the summation index and a P-recursive sequence and its shifts. There is a distinction between so-called normal and so-called special polynomials. Under the assumption that the corresponding difference field has no unnatural constants, we are able to predict the normal polynomials appearing in the denominator of a potential closed form. We can also handle the numerator. Our method is incomplete so far as we cannot predict the special polynomials appearing in the denominator. However, we do have some structural results about special polynomials for the setting under consideration.

math.CO↗

arXiv2024

Hilbert's Irreducibility Theorem for Linear Differential Operators

We prove a differential analogue of Hilbert's irreducibility theorem. Let $\mathcal{L}$ be a linear differential operator with coefficients in $C(\mathbb{X})(x)$ that is irreducible over $\overline{C(\mathbb{X})}(x)$, where $\mathbb{X}$ is an irreducible affine algebraic variety over an algebraically closed field $C$ of characteristic zero. We show that the set of $c\in \mathbb{X}(C)$ such that the specialized operator $\mathcal{L}^c$ of $\mathcal{L}$ remains irreducible over $C(x)$ is Zariski dense in $\mathbb{X}(C)$.

math.RA↗

arXiv2023

Stability Problems on D-finite Functions

This paper continues the studies of symbolic integration by focusing on the stability problems on D-finite functions. We introduce the notion of stability index in order to investigate the order growth of the differential operators satisfied by iterated integrals of D-finite functions and determine bounds and exact formula for stability indices of several special classes of differential operators. With the basic properties of stability index, we completely solve the stability problem on general hyperexponential functions.

cs.SC↗

arXiv2022

Galois Groups of Linear Difference-Differential Equations

We study the relation between the Galois group $G$ of a linear difference-differential system and two classes $\mathcal{C}_1$ and $\mathcal{C}_2$ of groups that are the Galois groups of the specializations of the linear difference equation and the linear differential equation in this system respectively. We show that almost all groups in $\mathcal{C}_1\cup \mathcal{C}_2$ are algebraic subgroups of $G$, and there is a nonempty subset of $\mathcal{C}_1$ and a nonempty subset of $\mathcal{C}_2$ such that $G$ is the product of any pair of groups from these two subsets. These results have potential application to the computation of the Galois group of a linear difference-differential system. We also give a criterion for testing linear dependence of elements in a simple difference-differential ring, which generalizes Kolchin's criterion for partial differential fields.

math.RA↗

arXiv2022

Differential Galois groups, specializations and Matzat's conjecture

We study families of linear differential equations parametrized by an algebraic variety $\mathcal{X}$ and show that the set of all points $x\in \mathcal{X}$, such that the differential Galois group at the generic fibre specializes to the differential Galois group at the fibre over $x$, is Zariski dense in $\mathcal{X}$. As an application, we prove Matzat's conjecture in full generality: The absolute differential Galois group of a one-variable function field over an algebraically closed field of characteristic zero is a free proalgebraic group.

math.AG↗

arXiv2021

Quasi-equivalence of heights in algebraic function fields of one variable

For points $(a,b)$ on an algebraic curve over a field $K$ with height $\mathfrak{h}$, the asymptotic relation between $\mathfrak{h}(a)$ and $\mathfrak{h}(b)$ has been extensively studied in diophantine geometry. When $K=\overline{k(t)}$ is the field of algebraic functions in $t$ over a field $k$ of characteristic zero, Eremenko in 1998 proved the following quasi-equivalence for an absolute logarithmic height $\mathfrak{h}$ in $K$: Given $P\in K[X,Y]$ irreducible over $K$ and $ε>0$, there is a constant $C$ only depending on $P$ and $ε$ such that for each $(a,b)\in K^2$ with $P(a,b)=0$, $$ (1-ε) °(P,Y) \mathfrak{h}(b)-C \leq °(P,X) \mathfrak{h}(a) \leq (1+ε) °(P,Y) \mathfrak{h}(b)+C. $$ In this article, we shall give an explicit bound for the constant $C$ in terms of the total degree of $P$, the height of $P$ and $ε$. This result is expected to have applications in some other areas such as symbolic computation of differential and difference equations.

cs.SC↗

arXiv2021

Separability Problems in Creative Telescoping

For given multivariate functions specified by algebraic, differential or difference equations, the separability problem is to decide whether they satisfy linear differential or difference equations in one variable. In this paper, we will explain how separability problems arise naturally in creative telescoping and present some criteria for testing the separability for several classes of special functions, including rational functions, hyperexponential functions, hypergeometric terms, and algebraic functions.

cs.SC↗

arXiv2021

Telescopers for differential forms with one parameter

Telescopers for a function are linear differential (resp. difference) operators annihilated by the definite integral (resp. definite sum) of this function. They play a key role in Wilf-Zeilberger theory and algorithms for computing them have been extensively studied in the past thirty years. In this paper, we introduce the notion of telescopers for differential forms with $D$-finite function coefficients. These telescopers appear in several areas of mathematics, for instance parametrized differential Galois theory and mirror symmetry. We give a sufficient and necessary condition for the existence of telescopers for a differential form and describe a method to compute them if they exist. Algorithms for verifying this condition are also given.

cs.SC↗

arXiv2020

Rational Solutions of First Order Algebraic Ordinary Differential Equations

Let $f(t, y,y')=\sum_{i=0}^d a_i(t, y)y'^i=0$ be a first order ordinary differential equation with polynomial coefficients. Eremenko in 1999 proved that there exists a constant $C$ such that every rational solution of $f(t, y,y')=0$ is of degree not greater than $C$. Examples show that this degree bound $C$ depends not only on the degrees of $f$ in $t,y,y'$ but also on the coefficients of $f$ viewed as polynomial in $t,y,y'$. In this paper, we show that if $$\max_{i=0}^d \{{\rm deg}(a_i,y)-2(d-i)\}>0 $$ then the degree bound $C$ only depends on the degrees of $f$, and furthermore we present an explicit expression for $C$ in terms of the degrees of $f$.

cs.SC↗

arXiv2020

Difference Galois groups under specialization

We present a difference analogue of a result given by Hrushovski on differential Galois groups under specialization. Let $k$ be an algebraically closed field of characteristic zero and $\mathbb{X}$ an irreducible affine algebraic variety over $k$. Consider the linear difference equation $$ σ(Y)=AY $$ where $A\in \mathrm{GL}_n(k(\mathbb{X})(x))$ and $σ$ is the shift operator $σ(x)=x+1$. Assume that the Galois group $G$ of the above equation over $\overline{k(\mathbb{X})}(x)$ is defined over $k(\mathbb{X})$ i.e. the vanishing ideal of $G$ is generated by a finite set $S\subset k(\mathbb{X})[X,1/\det(X)]$. For a ${\bf c}\in \mathbb{X}$, denote by $v_{\bf c}$ the map from $k[\mathbb{X}]$ to $k$ given by $v_{\bf c}(f)=f({\bf c})$ for any $f\in k[\mathbb{X}]$. We prove that the set of ${\bf c}\in \mathbb{X}$ satisfying that $v_{\bf c}(A)$ and $v_{\bf c}(S)$ are well-defined and the affine variety in $\mathrm{GL}_n(k)$ defined by $v_{\bf c}(S)$ is the Galois group of $σ(Y)=v_{\bf c}(A)Y$ over $k(x)$ is Zariski dense in $\mathbb{X}$. We apply our result to van der Put-Singer's conjecture which asserts that an algebraic subgroup $G$ of $\mathrm{GL}_n(k)$ is the Galois group of a linear difference equation over $k(x)$ if and only if the quotient $G/G^\circ$ by the identity component is cyclic. We show that if van der Put-Singer's conjecture is true for $k=\mathbb{C}$ then it will be true for any algebraically closed field $k$ of characteristic zero.

math.RA↗