arXiv subjects

Yangcheng Li

Publications and source records attributed to Yangcheng Li.

At least 19 recordsLinked to original sources

arXiv2026

Reconstruction of Binary Linear Systems and Profile Geometry of Sparse Krylov Strata

We study reconstruction and profile geometry for divisor schemes of binary linear systems and their sparse Krylov charts. Over the integers, the first nonzero equations of the complete embedded divisor scheme recover the defining linear system functorially under arbitrary base change, yielding a closed immersion from the Grassmannian of linear systems to the corresponding Hilbert scheme. For monomial systems in characteristic zero, arithmetic profiles classify the reduced factorization branches, determine their image dimensions and generic multiplicities, and control geometric reducedness. For complete progressions, the normalizations of the relation branches and their images are explicit products of projective spaces equipped with two natural polarizations. We give an affine normality criterion in terms of the associated Fourier data and a projective criterion obtained by adjoining an endpoint-allocation condition. These results place the closed sparse Krylov rank loci in a uniform reconstruction--normalization framework and yield explicit mixed-degree and formal-profile consequences. They also clarify the limit of normalization data alone: recovering a possibly nonnormal image algebra requires additional information not addressed here.

math.AG↗

arXiv2026

Schur--Plucker Geometry of the MDS Locus for Principal-Ideal Codes

Let $g$ be a monic polynomial of degree $r<n$, and let $C_g(n)$ be the coefficient-vector code formed by multiples $ug$ with $\deg(ug)<n$. We study the coefficient-space MDS locus $M_{n,r}$. The companion construction identifies coefficient space with the moduli of cyclic matrix-vector pairs, and the remainder-orbit map embeds it as a smooth complete intersection in the standard big cell of $\operatorname{Gr}(r,n)$. We prove that every normalized maximal Plucker coordinate pulls back, up to sign, to a power of the constant coefficient $A_0$ times a Schur polynomial $S_\kappa(g)=s_\kappa(\Lambda_g)$, where $\kappa\subseteq (n-r)^{r-1}$. Hence the universal MDS polynomial is \[D_{n,r}=A_0\prod_{\kappa\subseteq (n-r)^{r-1}}S_\kappa.\] This description yields a flat non-MDS boundary over $\mathbb{Z}$, explicit degree and finite-field estimates, and a length filtration governed by sparse multiples. It also gives bad-characteristic criteria on root-multiplicity strata and density-one results on the irreducible stratum. Finally, for $r\ge 3$ and $N\ge r+3$, every nonempty first-failure layer over an algebraically closed field has a dense open non-GRS locus.

cs.IT↗

arXiv2026

Cyclic Projective Orbits on Rational Normal Curves and MDS Codes

Let $A$ be a cyclic operator on an $r$-dimensional vector space over a field $k$, and let $z$ be a cyclic vector. Their Krylov code has parity-check matrix $(z,Az,\ldots,A^{n-1}z)$. For $r\ge 3$ and $n\ge r+3$, we prove that an MDS orbit segment lies on a rational normal curve precisely when the projective pair $(A,[z])$ is conjugate to one arising from the $(r-1)$-st symmetric-power action of $\mathrm{PGL}_2$. Over finite fields, for companion operators, this gives a complete classification of the generalized Reed--Solomon locus into split semisimple, two nonsplit semisimple, and unipotent families. Over an algebraically closed field $k$, the Zariski closure $\GRSsurf_{r,k}$ of the semisimple GRS coefficient locus is an irreducible rational surface, generically parameterized two-to-one by a two-dimensional torus of geometric-progression root sets; reversal is the generic ambiguity. The affine quotient of the parameter torus by reversal is the normalization of $\GRSsurf_{r,k}\cap D(a_0)$, its nonzero-constant-term open part. The codimension in the space of monic degree-$r$ polynomials is $r-2$. Frobenius descent gives an exact formula for the number of GRS polynomials over $\mathbb F_q$. A canonical remainder parity-check matrix defines the MDS locus by a principal open condition. For fixed $r\ge3$ and $n\ge r+3$, the proportion of all monic degree-$r$ polynomials over $\mathbb F_q$ whose companion codes are MDS and non-GRS tends to one as $q\to\infty$ through prime powers.

math.NT↗

arXiv2026

Cyclic Codes and Cyclically Covering Subspaces over Finite Fields

Let $q$ be a power of a prime $p$, and let $n$ be a positive integer. A subspace $U\subseteq \mathbb F_q^n$ is called cyclically covering if the union of all its cyclic shifts covers $\mathbb F_q^n$, and $h_q(n)$ denotes the maximum possible codimension of such a subspace. This paper studies cyclically covering subspaces via cyclic codes. We first prove that $h_q(n)=0$ if and only if every nonzero cyclic code in $\mathbb F_q^n$ contains a full-weight codeword. We also relate $h_q(n)$ to the maximum weights of cyclic codes. In particular, when $h_q(n)>0$, we obtain sharp bounds for the maximum weight of cyclic codes without full-weight codewords and provide explicit examples attaining these bounds. Moreover, we study the number of cyclic codes containing no full-weight codeword. We determine this number completely over $\mathbb F_2$, and give lower bounds over $\mathbb F_3$. From this, we prove that if $q\ge 3$ is an odd prime and $m\ge 4$ is an integer, then $h_q\left(\frac{q^m+1}{2}\right)>0$.

math.NT↗

arXiv2026

Discrete Fourier Transform Approach to Cyclically Covering Subspaces of $\mathbb{F}^n_q$

Let $q$ be a prime power and $n$ a positive integer. A subspace $ U \subseteq \mathbb{F}_q^n $ is called cyclically covering if the union of all its cyclic shifts covers the whole space $ \mathbb{F}_q^n $. Let $ h_q(n) $ denote the maximum possible codimension of such a subspace. When $\gcd(q,n)=1$, we derive necessary and sufficient conditions for $h_q(n)=0$ via Discrete Fourier Transforms, and prove this equality is equivalent to the existence of full-weight codewords in cyclic codes of $\mathbb{F}_q^n$. We also characterize codimension-$k$ cyclically covering subspaces. {Under suitable coprimality conditions on $m,n$ and on the multiplicative orders of $q$, we prove that the vanishing of $h_q(m)$ and $h_q(n)$ is preserved under their product.} Based on these results, we give a unified characterization of $h_q(n)$ in the case where $q$ and $n$ are primes with $n>q$ and $q$ being a primitive root modulo $n$. Specifically, $h_2(n) \geq 2$ and $h_q(n) = 0$ for $q \neq 2$. We prove that $h_3(n) \ge 1$ for every prime $n > 3$ with odd $\operatorname{ord}_n(3)$. Moreover, for any prime $q > 3$, the Generalized Riemann Hypothesis implies the existence of infinitely many primes $n > q$ such that $q$ is not a primitive root modulo $n$ and $h_q(n) = 0$. We provide algebraic interpretations for the inequalities $h_q(mn)\ge\max\{h_q(m),h_q(n)\}$ and $h_q(mn)\ge h_q(m)+h_q(n)$. Using Galois descent, we prove $h_{q^m}(n)\le h_q(n)$. Furthermore, we generalize a class of constructions that achieve the upper bound $\lfloor\log_q(n)\rfloor$. Finally, under the Generalized Riemann Hypothesis, we obtain average lower bounds of $h_q(n)$ for $q=2,3$.

math.NT↗

arXiv2026

The Arithmetic Geometry of Square-Sided Heron Triangles

We study rational Heron triangles with two marked square sides using elliptic curves and K3 surfaces. An explicit quartic-to-elliptic correspondence parametrizes marked similarity classes by rational points satisfying a positivity condition, modulo $(x,y)\sim(x,-y)$. We determine the generic Mordell--Weil group, prove that every $k\in\mathbf Q\setminus\{0,\pm1\}$ supports infinitely many scalene classes with exactly two square sides, and construct a primitive family with $N(X)\gg X^{1/4}$. Requiring the third side to be square gives a genus-three Ciani quartic whose Jacobian is $\mathbf Q$-isogenous to a product of three elliptic curves. Geometrically, the two constructions give inequivalent elliptic fibrations on a single singular K3 surface, with geometric Mordell--Weil ranks $2$ and $0$. The minimal resolution of the all-square locus is a surface of general type with invariants $(K^2,p_g,q)=(2,3,0)$. Assuming weak Bombieri--Lang, parameters yielding a nondegenerate all-square triangle form a thin subset of $\mathbf P^1(\mathbf Q)$.

math.NT↗

arXiv2026

RedFuser: An Automatic Operator Fusion Framework for Cascaded Reductions on AI Accelerators

Operator fusion, as a key performance optimization technique in the deployment of AI models, significantly improves execution efficiency and has been widely adopted in modern AI compilers. However, for cascaded reduction operations involving multiple loops with inter-loop data dependencies, such as the safe softmax followed by GEMM within attention mechanisms, existing compilers lack effective automated fusion and kernel generation capabilities. Although some works have addressed specific instances through hand-crafted fusion strategies, their solutions are limited in generality and difficult to extend to other similar structures. Given the prevalence of such computational patterns in deep learning models, there remains significant untapped potential in achieving general and automated fusion optimization. In this paper, we present a formal theoretical methodology for analyzing cascaded reductions which can fuse them into a single loop and introduce an incremental computation form. Based on this methodology, we design Reduction Fuser (RedFuser), a framework that automatically identifies supported cascaded reduction patterns and generates optimized fused kernels. Experiments show that RedFuser successfully fuses diverse workloads, achieving up to 2$\times$ to 5$\times$ speedup over state-of-the-art AI compilers and matching the performance of highly optimized hand-written kernels. The code is available at https://github.com/alibaba/redfuser

cs.AR↗

arXiv2026

On cyclically covering subspaces of $\mathbb{F}^n_q$

For a prime power $ q $ and a positive integer $ n $, a subspace $ U \subseteq \mathbb{F}_q^n $ is called cyclically covering if the union of all its cyclic shifts covers the whole space $ \mathbb{F}_q^n $. Let $ h_q(n) $ denote the maximum possible codimension of such a subspace. This paper focuses on the case $ h_q(n) = 0 $. We provide necessary and sufficient conditions under which $ h_q(n) = 0 $ holds. As an application, we show that $ h_q(\ell^t) = 0 $ whenever $ q $ is a primitive root modulo $ \ell^t $. Moreover, we prove that if $ n $ is odd and $ h_q(n) = 0 $, then also $ h_q(2n) = 0 $. As an example, we show that $ h_3(11) =h_3(16) = 1 $. Furthermore, we investigate the relationship between the coverings of $\mathbb{F}_{q^m}^n$ and $\mathbb{F}_q^{mn}$, and obtain several sufficient conditions for $h_{q^m}(n) = 0$. Specifically, we derive that if $n = 3$ or $n = 2^d$ (where $d$ is a nonnegative integer), then $h_4(n) = 0$.

math.NT↗

arXiv2026

An Infinite Family of Primitive Heron Triangles with Two Sides as Perfect Squares

A primitive Heron triangle is a triangle with integral sides and integral area where the greatest common divisor of the lengths of its sides is $1$. By utilizing the theory of elliptic curves, we prove that there exist infinitely many primitive Heron triangles with two sides being perfect squares. In this process, we nest one elliptic curve into another and find a surprising rational point. All the Heron triangles corresponding to this rational point are primitive. This result would imply the possible existence of infinitely many primitive Heron triangles with all three sides being perfect squares.

math.NT↗

arXiv2025

Determination of Some Types of Permutations over $\mathbb{F}_q^2$ with Low-Degree

The characterization of permutations over finite fields is an important topic in number theory with a long-standing history. This paper presents a systematic investigation of low-degree bivariate polynomial systems $F=(f_1(x,y),f_2(x,y))$ defined over $\mathbb{F}_{q}^2$. Specifically, we employ Hermite's Criterion to completely classify bivariate quadratic permutation polynomial systems, while utilizing the theory of permutation rational functions to give a full classification of bivariate 3-homogeneous permutation polynomial systems. Furthermore, as an application of our findings, we provide an explicit characterization of the permutation binomials of the form $x^3+ax^{2q+1}$ over $\mathbb{F}_{q^2}$ with characteristic $p\neq3$, thereby resolving a significant special case within this classical research domain.

math.NT↗

arXiv2025

Permutation polynomials of the form $x+\gamma \mathrm{Tr}(H(x))$

Given a polynomial $ H(x) $ over $\mathbb{F}_{q^n}$, we study permutation polynomials of the form $ x + \gamma \mathrm{Tr}(H(x)) $ over $\mathbb{F}_{q^n}$. Let \[P_H=\{\gamma\in \mathbb{F}_{q^n} : x+\gamma \mathrm{Tr}(H(x))~\text{is a permutation polynomial}\}.\] We present some properties of the set $P_H$, particularly its relationship with linear translators. Moreover, we obtain an effective upper bound for the cardinality of the set $P_H$ and show that the upper bound can reach up to $q^n - q^{n - 1}$. Furthermore, we prove that when the cardinality of the set $P_H$ reaches this upper bound, the function $\mathrm{Tr}(H(x))$ must be an $\mathbb{F}_q$-linear function. Finally, we study two classes of functions $H(x)$ over $\mathbb{F}_{q^2}$ and determine the corresponding sets $P_H$. The sizes of these sets $P_H$ are all relatively small, even only including the trivial case.

math.NT↗

arXiv2025

$(G,F)$-points on $\mathbb{Q}$-algebraic varieties

Let $G\in \mathbb{Q}[x,y,z]$ be a polynomial, and let $V(G)$ be the $\mathbb{Q}$-algebraic variety corresponding to $G$, i.e., $V(G)=\{P\in\mathbb{Q}^3~|~G(P)=0\}$. Let \[\begin{split} F:\quad &\mathbb{Q}^3\rightarrow \mathbb{Q}^3,\\ &(x,y,z)\mapsto (f(x),f(y),f(z)) \end{split}\] be a vector function, where $f\in \mathbb{Q}[x]$. It is easy to know that the function obtained by the composition of $G$ and $F$, denoted as $G\circ F$, is still in $\mathbb{Q}[x,y,z]$. Moreover, let $V(G\circ F)$ be the $\mathbb{Q}$-algebraic variety corresponding to $G\circ F$, i.e., $V(G\circ F)=\{P\in\mathbb{Q}^3~|~G\circ F(P)=0\}$. A rational point $P$ is called a $(G,F)$-point on $V(G)$ if $P$ belongs to the intersection of $V(G)$ and $V(G\circ F)$, that is $P\in V(G)\cap V(G\circ F)$. Denote $\langle G,F\rangle$ as the set consisting of all $(G,F)$-points on $V(G)$. Obviously, $\langle G,F\rangle$ is a $\mathbb{Q}$-algebraic variety. In this paper, we consider the algebraic variety $\langle G,F\rangle$ for some specific functions $G$ and $F$. For these specific functions $G$ and $F$, we prove that $\langle G,F\rangle$ will be isomorphic to a certain elliptic curve. We also analyze some properties of these elliptic curves.

math.NT↗

arXiv2023

A new perspective of arithmetic billiards

We study the problem of arithmetic billiards from a new perspective. We first raise a similar problem about reflecting lights inside grids. For the solution to this problem, we will give three proofs. Next, we consider a similar problem in plane grids and give its solution. Moreover, we extend this two problems to $p$-dimensional space, where $p\geq2$. In this process, we introduce two mappings about finite discrete sets, and get two finite abelian groups. In addition, we give the definition of circular sequences and consider some combinatorial properties of circular sequences.

math.NT↗

arXiv2023

On the Diophantine system involving pairs of triangles with the same area and the same perimeter

Many authors studied the problem that rational triangle pairs (triangle-parallelogram pairs) with the same area and the same perimeter. They investigated this problem by solving the rational solutions of the corresponding Diophantine equations. In this paper, we give a unified description of this problem by using the affine transformation in a rectangular coordinate system. According to the fact that two triangles with the same area and the same perimeter determine an affine transformation, this problem can be reduced to solving a specific Diophantine system. Moreover, we will give some rational solutions to this Diophantine system.

math.NT↗

arXiv2017

Whispering gallery mode hybridization in photonic molecules

This work takes inspiration from chemistry where the spectral characteristics of the molecules are determined by hybridization of electronic states evolving from the individual atomic orbitals. Based on analogy between quantum mechanics and the classical electrodynamics, we sorted dielectric microspheres with almost identical positions of their whispering gallery mode (WGM) resonances. Using these microspheres as classical photonic atoms, we assembled them in a wide range of structures including linear chains and planar photonic molecules. We studied WGM hybridization effects in such structures using side coupling by tapered microfibers as well as finite difference time domain modeling. We demonstrated that the patterns of WGM spectral splitting are representative of the symmetry, number of constituting atoms and topology of the photonic molecules which in principle can be viewed as "spectral signatures" of various molecules. We also show new ways of controlling WGM coupling constants in such molecules. Excellent agreement was found between measured transmission spectra and spectral signatures of photonic molecules predicted by simulation.

physics.optics↗

arXiv2015

Microspherical photonics: Giant resonant light forces, spectrally resolved optical manipulation, and coupled modes of microcavity arrays

In this dissertation novel resonant propulsion of dielectric microspheres is studied with the goal of sorting spheres with identical resonances, which are critical for developing microspherical photonics. First, evanescent field couplers were developed by fixing tapered microfibers in mechanically robust platforms. The tapers were obtained by chemical etching techniques. Using these platforms, WGMs modal numbers, coupling regimes and quality factors were determined for various spheres and compared with theory. Second, the spectroscopic properties of photonic molecules formed by spheres with better than 0.05% uniformity of WGM resonances were studied. It was shown that various spatial configurations of coupled-cavities present relatively stable mode splitting patterns in the fiber transmission spectra which can be used as spectral signatures to distinguish such photonic molecules. The third part is the study of giant resonant propulsion forces exerted on microspheres. This effect was observed in suspensions of polystyrene spheres with sufficiently large diameters. By integrating optical tweezers for individual sphere manipulation, the wavelength detuning between a tunable laser and WGMs in each of the spheres was precisely controlled. Resonant enhancement of optical forces was directly demonstrated in experiments. The spectral shape, position and magnitude of the observed propulsion force peaks were explained by efficient transfer of light momentum to microspheres under resonant conditions. The peak magnitude of the resonant force is shown to approach total absorption limit imposed by the conservation of momentum. The transverse movement of the spheres during the propulsion process was studied and the existence of a stable radial trap was demonstrated. Giant resonant propulsion forces can be used for large-scale sorting of microspheres with ultrahigh uniform resonant properties.

physics.optics↗

arXiv2015

Movable thin films with embedded high-index microspheres for super-resolution microscopy

Microsphere-assisted imaging emerged as a surprisingly simple way of achieving optical super-resolution imaging. In this work, we use movable PDMS thin films with embedded high-index barium titanate glass microspheres a sample scanning capability was developed, thus removing the main limitation of this technology based on its small field-of-view.

physics.optics↗

arXiv2015

Spectrally resolved resonant propulsion of dielectric microspheres

Use of resonant light forces opens up a unique approach to high-volume sorting of microspherical resonators with much higher uniformity of resonances compared to that in coupled-cavity structures obtained by the best semiconductor technologies. In this work, the spectral response of the propulsion forces exerted on polystyrene microspheres near tapered microfibers is directly observed. The measurements are based on the control of the detuning between the tunable laser and internal resonances in each sphere with accuracy higher than the width of the resonances. The measured spectral shape of the propulsion forces correlates well with the whispering-gallery mode resonances in the microspheres. The existence of a stable radial trap for the microspheres propelled along the taper is demonstrated. The giant force peaks observed for 20-μm spheres are found to be in a good agreement with a model calculation demonstrating an efficient use of the light momentum for propelling the microspheres.

physics.optics↗