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Xiping Zhang

Publications and source records attributed to Xiping Zhang.

12 recordsLinked to original sources
arXiv2026

Danus: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory

Recent LLM-based mathematical reasoning agents have begun to tackle research-level problems and, in several cases, have contributed to the resolution of open problems. However, scaling and orchestrating such agents effectively remains challenging, due to the difficulty of coordinating parallel proof search while keeping intermediate claims organized and reliable. In this paper, we propose Danus, an orchestration system for research-level mathematical reasoning centered on a shared fact graph as a global memory-management mechanism. Danus consists of a main agent that performs planning and coordination, multiple worker agents that carry out proof search in parallel, and a stateless verifier that checks proposed mathematical claims before they are admitted into the fact graph. Each verified fact is stored together with its proof and logical dependencies, allowing the system to build long arguments incrementally while keeping the shared proof state organized. The main agent periodically summarizes the evolving proof state, redirects workers across promising directions, and supports interaction with human mathematicians through progress reports. We evaluate Danus through six research-level case studies in algebraic geometry, singularity theory, and combinatorics, illustrating how the fact-graph memory mechanism enables Danus to construct long, detailed mathematical proofs. Our results suggest that fact-graph-based orchestration provides an effective route toward scaling mathematical reasoning agents for long-horizon research problems. Danus is open source at https://github.com/frenzymath/Danus.

cs.AI↗
arXiv2026

Criteria of isolated weighted homogeneous hypersurface singularities using Logarithmic vector fields

We prove a conjecture of da Silva Machado and Seade that characterizes weighted homogeneous isolated hypersurface singularities through the existence of a logarithmic vector field transverse to the link. For a reduced isolated hypersurface germ $(D,0)$ in $\C^{n+1}$ with $n\ge2$, or with $n=1$ and $D$ irreducible, we prove that weighted homogeneity is equivalent to the existence, in suitable coordinates, of a logarithmic vector field everywhere transverse in the real-Euclidean sense to all small links. We also prove the equivalent formulation that $(D,0)$ admits an ambient holomorphic vector field tangent to $D$ that has a non-degenerate isolated singularity at $0$. We further show that the transversality condition must be read after allowing a coordinate change: there exists a weighted homogeneous germ admitting no logarithmic field transverse to the standard round links in certain linear coordinates. The main result of this paper was obtained by the Rethlas system.

math.AG↗
arXiv2026

Asymptotic Betti bounds for hypersurfaces in a singular variety

We show that for any degree $d$ hypersurface $Y \subset X$ in a possibly singular projective variety $X \subset \mathbf{P}^N$, the total Betti number of $Y$ is bounded by $3\text{deg}(X)\cdot d^n + C\cdot d^{n-1}$ for some explicit constant $C > 0$ independent of $d$ and $Y$. When $X$ is a local complete intersection, the bound improves to $\text{deg}(X)\cdot d^n + C\cdot d^{n-1}$. In this case, the bound is asymptotically sharp. Similar bounds are also established for general constructible sheaves.

math.AG↗
arXiv2025

Microlocal indices and Chern Classes of Foliations

In this paper, we study how global index formulas arise in the theory of one-dimensional holomorphic foliation from the microlocal point of view. We give short proofs and generalizations to a few exisiting index formulas concerning Schwartz, GSV and logarithmic indices.

math.AG↗
arXiv2025

A Syzygy Rank Characterization of Strongly Euler Homogeneity for Projective Hypersurfaces

In this paper we give a characterization of strongly Euler homogeneous singular points on a reduced complex projective hypersurface $D=V(f)\subset \PP^n$ using the Jacobian syzygies of $f$. The characterization compares the ranks of the first syzygy matrices of the global Jacobian ideal $J_f$ and its quotient $J_f/(f)$. When $D$ has only isolated singularities, our characterization refines a recent result of Andrade-Beorchia-Dimca-Mir\'{o}-Roig. We also prove a generalization of this characterization to smooth projective toric varieties.

math.AG↗
arXiv2025

Logrithmic Versions of Ginzburg's Sharp Operation for Free Divisors

Let $M$ be a complex manifold, $D\subset M$ a free divisor and $U=M\setminus D$ its complement. In this paper we study the characteristic cycle $\textup{CC}(\gamma\cdot \ind_U)$ of the restriction of a constructible function $\gamma$ on $U$. We globalise Ginzburg's local sharp construction and introduce the log transversality condition, which is a new transversality condition about the relative position of $\gamma$ and $D$. We prove that the log transversality condition is satisfied if either $D$ is normal crossing and $\gamma$ is arbitrary, or $D$ is holonomic strongly Euler homogheneous and $\gamma$ is non-characteristic. Under the log transversality assumption we establish a logarithmic pullback formula for $\textup{CC}(\gamma\cdot \ind_U)$. Mixing Ginzburg's sharp construction with the logarithmic pullback, we obtain a double restriction formula for the Chern-Schwartz-MacPherson class $c_*(\gamma\cdot \ind_{D\cup V})$ where $V$ is any reduced hypersurface in $M$. Applications of our results include the non-negativity of Euler characteristics of effective constructible functions, and CSM classes of hypersurfaces in the open manifold $\mathbb{P}^n\setminus D$ when $D$ is a linear free divisor or a free hyperplane arrangement.

math.AG↗
arXiv2021

Geometric Invariants of Recursive Group Orbit Stratification

The local Euler obstructions and the Euler characteristics of linear sections with all hyperplanes on a stratified projective variety are key geometric invariants in the study of singularity theory. Despite their importance, in general it is very hard to compute them. In this paper we consider a special type of singularity: the recursive group orbits. They are the group orbits of a sequence of $G_n$ representations $V_n$ satisfying certain assumptions. We introduce a new intrinsic invariant called the $c_{sm}$ invariant, and use it to give explicit formulas to the local Euler obstructions and the sectional Euler characteristics of such orbits. In particular, the matrix rank loci are examples of recursive group orbits. Thus as applications, we explicitly compute these geometry invariants for ordinary, skew-symmetric and symmetric rank loci. Our method is systematic and algebraic, thus works for algebraically closed field of characteristic $0$. Moreover, in the complex setting we also compute the stalk Euler characteristics of the Intersection Cohomology Sheaf complexes for all three types of rank loci.

math.AG↗
arXiv2021

Characteristic Classes of Homogeneous Essential Isolated Determinantal Varieties

The (homogeneous) Essentially Isolated Determinantal Variety is the natural generalization of generic determinantal variety, and is fundamental example to study non-isolated singularities. In this paper we study the characteristic classes on these varieties. We give explicit formulas of their Chern-Schwartz-MacPherson classes and Chern-Mather classes via standard Schubert calculus. As corollaries we obtain formulas for their (generic) sectional Euler characteristics, characteristic cycles and polar classes.

math.AG↗
arXiv2021

Local Euler Obstructions of Reflective Projective Varieties

In this note we introduce the concept of reflective projective varieties. These are stratified projective varieties with certain dimension constraints on their dual varieties. We prove that for such varieties, the Chern-Schwartz-MacPherson classes of the strata completely determine the local Euler obstructions and the polar degrees. We also propose an algorithm to compute the local Euler obstructions when such varieties are formed by group orbits. As examples we compute the local Euler obstructions of quadratic hypersurfaces and ordinary determinantal varieties to illustrate our method.

math.AG↗
arXiv2017

Chern classes and Characteristic Cycles of Determinantal Varieties

Let $K$ be an algebraically closed field of characteristic $0$. For $m\geq n$, we define $τ_{m,n,k}$ to be the set of $m\times n$ matrices over $K$ with kernel dimension $\geq k$. This is a projective subvariety of $\bbP^{mn-1}$, and is called the (generic) determinantal variety. In most cases $τ_{m,n,k}$ is singular with singular locus $τ_{m,n,k+1}$. In this paper we give explicit formulas computing the Chern-Mather class ($c_M$) and the Chern-Schwartz-MacPherson class ($c_{SM}$) of $τ_{m,n,k}$, as classes in the projective space. We also obtain formulas for the conormal cycles and the characteristic cycles of these varieties, and for their generic Euclidean Distance degree. Further, when $K=\bbC$, we prove that the characteristic cycle of the intersection cohomology sheaf of a determinantal variety agrees with its conormal cycle (and hence is irreducible). Our formulas are based on calculations of degrees of certain Chern classes of the universal bundles over the Grassmannian. For some small values of $m,n,k$, we use Macaulay2 to exhibit examples of the Chern-Mather classes, the Chern-Schwartz-MacPherson classes and the classes of characteristic cycles of $τ_{m,n,k}$. On the basis of explicit computations in low dimensions, we formulate conjectures concerning the effectivity of the classes and the vanishing of specific terms in the Chern-Schwartz-MacPherson classes of the largest strata $τ_{m,n,k}\smallsetminus τ_{m,n,k+1}$. The irreducibility of the characteristic cycle of the intersection cohomology sheaf follows from the Kashiwara-Dubson's microlocal index theorem, a study of the `Tjurina transform' of $τ_{m,n,k}$, and the recent computation of the local Euler obstruction of $τ_{m,n,k}$ .

math.AG↗
arXiv2017

Local Euler Obstruction and Chern-Mather classes of Determinantal Varieties

For $m\geq n$, Let $K$ be an algebraic closed base field, and define $τ_{m,n,k}$ to be the set of $m\times n$ matrices over $K$ with kernel dimension $\geq k$. This is a projective subvariety of $\mathbb{P}^{mn-1}$, and is usually called determinantal variety. In most cases $τ_{m,n,k}$ is singular with singular locus $τ_{m,n,k+1}$. In this paper we compute the local Euler obstruction of $τ_{m,n,k}$, and we prove that the characteristic cycle of the intersection cohomology complex of $τ_{m,n,k}$ is irreducible. We also give an explicit formula for the Chern-Mather class of $τ_{m,n,k}$ as a class in projective space. The irreducibility of the intersection cohomology characteristic cycle follows from the explicit computation of the local Euler obstruction, a study of the `Tjurina transforms' of determinantal varieties, and the Kashiwara-Dubson's microlocal index theorem. Our explicit formulas are based on calculations of degrees of certain Chern classes of the universal bundles over the Grassmannian. We use The Schubert 2 package in Macaulay2 to exhibit examples of the Chern-Mather class and the class of the characteristic cycle of $τ_{m,n,k}$ for some small values of $m,n,k$. Over the complex numbers, the local Euler obstruction of $τ_{m,n,k}$ was recently computed by N.~Grulha, T.~Gaffney and M.~Ruas by methods in complex geometry.

math.AG↗