arXiv subjects

Yaoran Yang

Publications and source records attributed to Yaoran Yang.

At least 19 recordsLinked to original sources

arXiv2026

The 196560 auxiliary-function conjecture for the Leech lattice

Cohn and Kumar conjectured in 2009 that there is a radial Schwartz function $g\colon\R^{24}\to\R$ satisfying $g(r)\leq0$ for $r\geq\sqrt6$, $\widehat g(r)\geq0$ for $r\geq0$, $g(2)>0$, and $(\widehat g(0)-g(0))/g(2)=196560$. We construct such functions from the radial Fourier interpolation basis in dimension $24$. If $a_2,b_2$ denote the basis functions dual to value and radial-derivative interpolation at radius $2$, then $g_C=a_2-Cb_2$ has exactly the nodal data needed for Poisson summation over the Leech lattice. The sphere-packing magic function identifies $b_2$ and supplies its strict signs. We prove that the removable quotients $a_2/b_2$ and $\widehat a_2/\widehat b_2$ are bounded on the required half-lines. The noncompact step follows from exact coefficient extraction in the interpolation kernel and an $S$-cusp expansion. Both quotients tend to $(43+240\log2)/15$; for all sufficiently large radii they lie on opposite sides of this limit, with an explicit first exponential correction. Consequently the admissible parameters in this affine family form a nonempty closed ray, and every member proves the conjectured identity. The same interpolation basis recovers every nontrivial Leech-shell coefficient.

math.NT↗

arXiv2026

Frobenius-orbit slicing and uniform elimination of positive-dimensional singular loci

Let $\cX\subseteq\Pj^n_{\Z}$ be a fixed integral quasiprojective subscheme, smooth over $\Z$ of relative dimension $r$. For each fixed $m\ge1$, we bound the probability that the $m$th principal-parts jet of the restriction of a uniform degree-$d$ form to $\cX_p$ has a positive-dimensional zero scheme. The bound is $C(d+1)^{N_m}p^{-\lambda_m(d)}$, where $N_m=\binom{r+m}{m}$ and $\lambda_m(d)=\floor{m(d+1)/(m+1)}$. For $m=1$, this gives the Bertini singular-locus estimate $C(d+1)^{r+1}p^{-\ceil{d/2}}$. It settles Poonen's arithmetic Bertini Conjecture~5.2 and, after increasing the degree threshold, yields $p^{-A}$ for every fixed $A>0$. For $c\le r$ independent hypersurfaces, the probability of a positive-dimensional Jacobian rank-degeneracy locus is bounded both by $C\sum_i(d_i+1)^{r+1}p^{-\ceil{d_i/2}}$ and by $C'(d_{\min}+1)^{r+1}p^{-\ceil{d_{\min}/2}}$. The proof uses filtered $Q$-adic decompositions, triangular normal Taylor blocks, and Jacobian-pivot charts of uniformly controlled complexity.

math.AG↗

arXiv2026

Counterexamples to Charpin's Conjecture on BCH codes

We construct an infinite family of $q$-ary primitive narrow-sense BCH codes whose minimum distance strictly exceeds the Bose distance; in fact, the gap between the two can be arbitrarily large as the length of the code tends to infinity. The key idea is to embed these BCH codes in a suitably large punctured generalized Reed--Muller code, whose codeword weights obey divisibility conditions supplied by Ax's theorem. This divisibility forces the minimum distance of the BCH codes far above the Bose distance. In particular, our family disproves a longstanding conjecture of Charpin asserting that this difference is at most four.

cs.IT↗

arXiv2026

Exact Minimum Distance of the Ding--Li--Xia Cyclic Codes

The cyclic codes $\mho(q,m,h)$ introduced by Ding, Li, and Xia form a nonbinary generalization of punctured binary Reed--Muller codes. Ding, Li, and Xia established the bounds $(q^{h+1}-1)/(q-1)\leq d(\mho(q,m,h))\leq 2q^h-1$ and asked whether the BCH lower bound is always exact. This paper proves that, for every prime power $q$, every $m\geq 2$, and every $1\leq h\leq m-1$, the minimum distance is $d(\mho(q,m,h))=(q^{h+1}-1)/(q-1)$. The upper bound is obtained by an explicit projective-subspace construction. For any $(h+1)$-dimensional $\F_q$-subspace $V$ of $\F_{q^m}$, the set $V^{[q-1]}=\{x^{q-1}:x\in V\setminus\{0\}\}$ supports a codeword of weight $(q^{h+1}-1)/(q-1)$. Its membership in $\mho(q,m,h)$ follows from a vanishing lemma for subspace power sums and the digit-sum estimate $s_q((q-1)a)\leq(q-1)\wtq(a)$. The constructed codeword meets the BCH lower bound and therefore determines the exact minimum distance.

cs.IT↗

arXiv2026

Hyperplane anti-Bertini embeddings over finite fields

Baker asked, as recorded by Poonen, whether a fixed smooth quasiprojective variety over a finite field must have a smooth rational hyperplane section after every sufficiently high-dimensional linearly nondegenerate embedding. Poonen predicted a negative answer for every positive-dimensional variety. We prove this predicted negative answer for each prescribed variety: if $X$ is nonempty, smooth, quasiprojective, and of pure positive dimension over $\F_q$, then for every sufficiently large $N$ there is a locally closed embedding $X\hookrightarrow\PP^N_{\F_q}$ whose components remain linearly nondegenerate after arbitrary scalar extension, but whose every $\F_q$-rational hyperplane section is singular. The construction assigns one closed point of $X$ to each rational hyperplane and forces the pulled-back linear form to have zero first-order jet at that point.

math.AG↗

arXiv2026

A Proof of a Conjecture of Zhi-Wei Sun on a Truncated Legendre-Symbol Determinant

Let $p\ge7$ be a prime with $p\equiv3\pmod4$ and let $\chi$ be the Legendre symbol modulo $p$. We prove that $\det[x+\chi(j-k)]_{0\le j,k\le(p-7)/2}=\floor{(p-2)/3}^{2}x$ in $\mathbb{C}[x]$, which settles Conjecture~3.4 of Zhi-Wei Sun. The truncated matrix is a corner of Chapman's Legendre-symbol matrix $C$, and its determinant can be expressed through a handful of entries of $C^{-1}$ and of $C^{-1}\one$. Those entries are in turn read off from Vsemirnov's cyclotomic factorization of $C$ with the help of Schur's Pfaffian identity.

math.NT↗

arXiv2026

Finite and Urysohn obstructions to Sabok's S-prime simplex questions

Sabok asked whether the compact convex set $S'(X)$ attached to a separable metric space of diameter at most one is always a simplex, and whether $S'(\mathbb U_1)$ is the Poulsen simplex. We give negative answers. For finite $X=\{x_1,\ldots,x_m\}$, $S'(X)$ is affinely homeomorphic to the convex hull of the rows $r_i=(d(x_i,x_1),\ldots,d(x_i,x_m))$ of the distance matrix; it is a simplex exactly when these rows are affinely independent. The diameter-one four-cycle gives the minimal finite obstruction. For the Urysohn sphere, using the rational Urysohn sphere $D$ as coordinates, we identify the coordinate model $S'_D(\mathbb U_1)$ with the Kat\'etov compactum $K(D)$. Four explicit extreme points $f_A,g_A,\mathbf 1,\mathbf h$ satisfy $f_A+g_A=\mathbf 1+\mathbf h$, giving two distinct representing measures for $(3/4)\mathbf 1$. Hence $S'(\mathbb U_1)$ is not a Choquet simplex.

math.MG↗

arXiv2026

The least non-partitionable zero-sum subset for zero-sum triples in finite abelian groups

Let $G$ be a finite abelian group, and let $\mu(G)$ denote the least size of a subset $S\subseteq G$ with $3\mid |S|$, total sum zero, and no partition into zero-sum triples; put $\mu(G)=\infty$ if no such subset exists. We prove the exact classification $\mu(G)=\infty$ precisely for groups of order at most $8$ and for $G\cong C_3^2$, while $\mu(G)=6$ for every other finite abelian group. The cyclic special case gives $\mu(\mathbb Z_n)=\infty$ for $1\le n\le 8$ and $\mu(\mathbb Z_n)=6$ for $n\ge 9$, answering the corresponding non-partite cyclic question. We also record a higher-uniformity interval construction which explains the large cyclic witnesses as the case $k=3$ of a general $k$-tuple obstruction.

math.CO↗

arXiv2026

Excess Obstructions and Layer-Contained Certificates for the Hypergraph Nash--Williams--Tutte Conjecture

Guo, Li, Shangguan, Tamo, and Wootters proposed a hypergraph analogue of the Nash--Williams--Tutte theorem, asserting that every $k$-weakly-partition-connected hypergraph admits a $k$-distinguishable tree assignment. We identify a sharp edge-count obstruction to the literal statement. A full tree assignment has $\rho(H)=\sum_{e\in E(H)}(|e|-1)$ labelled graph edges, whereas an ordered decomposition into $k$ spanning trees has exactly $k(t-1)$ edges. Since weak partition connectivity implies only $\rho(H)\ge k(t-1)$, every strict inequality rules out a full decomposition. In particular, for all $t\ge2$, $k\ge1$, and $q\ge1$, the hypergraph consisting of $k+q$ labelled copies of the full hyperedge is $k$-weakly-partition-connected but admits no $k$-distinguishable full tree assignment. We therefore isolate the critical regime $\rho(H)=k(t-1)$ and prove that every tree assignment of a critical $k$-weakly-partition-connected hypergraph admits a full ordered decomposition into $k$ spanning trees. Consequently, the critical conjecture reduces to the existence of some tree assignment having a decomposition whose signature fiber is a singleton. We establish this property for layer-contained certificates, without a star hypothesis or a rank restriction in interior layers. These certificates also imply weak partition connectivity by a quotient-rank argument and are stable under one-vertex sums. Finally, we derive a decomposition-indexed generalized-Laplace expansion for the relevant intersection-matrix minor, together with an exact formula for its collected monomial coefficients; we also give the required row-and-column perfect-shuffle normalization, identify criticality with the square full-assignment row count, and separate the critical conjecture from the additional overfull pruning problem.

math.CO↗

arXiv2026

A Uniform Random-Lattice Tail Bound for the SVP Kissing-Profile Parameter

A recent SICOMP paper on classical and quantum algorithms for the shortest vector problem introduced a lattice-dependent parameter $\gamma(L)$, bounded universally in the exponential sense by $2^{0.402n+o(n)}$, and conjectured that this parameter is $2^{o(n)}$ for most lattices. We prove the Haar--Siegel random-lattice version in a stronger, dimension-uniform form. Let $X_n=\operatorname{SL}_n(\R)/\operatorname{SL}_n(\Z)$, let $\mu_n$ be its invariant probability measure, and let $\gamma(L)=\sup_{r\ge1} N_L(r\lambda_1(L))/r^n$, where $N_L(R)$ counts nonzero vectors of $L$ of Euclidean norm at most $R$. For every $n\ge3$ and every $T>0$, \[ \mu_n\{L\in X_n:\gamma(L)>T\}\le C T^{-1} \] with an absolute constant $C$. Consequently, for every sequence $a_n\to\infty$, $\gamma(L_n)\le a_n$ with $\mu_n$-probability tending to one; in particular $\gamma(L_n)=2^{o(n)}$ with high probability. In the product model of independent Haar--Siegel lattices, $\gamma(L_n)\le \exp(\sqrt n)$ eventually almost surely. The proof uses Rogers's second-moment estimate only through a dyadic self-normalization argument around the random scale $\lambda_1(L)$.

math.NT↗

arXiv2026

Betti Numbers of Cut Complexes of Squared Paths and a Recurrence Conjecture

For a graph $G$ on $[n]$, the $k$-cut complex $\Delta_k(G)$ has facets $[n]\setminus T$, where $T$ ranges over the $k$-subsets that induce disconnected subgraphs of $G$. Bayer, Denker, Jeli\'c Milutinovi\'c, Sundaram, and Xue proved that $\Delta_k(P_n^2)$ is shellable for $n\ge k+3$ and conjectured a finite-difference recurrence for its top reduced Betti number along each diagonal $n-k=r$. We prove the exact formula \[ \beta(k,n)=\binom{n-1}{k-1} -\sum_{j=0}^{\min\{k-1,n-k\}}\binom{k-1}{j}(n-k-j+1)+(n-k) \] for $k\ge2$ and $n\ge k+3$. The proof rests on a complete classification of complements of cardinality at least $k$ that contain no disconnected $k$-subset: every such complement has size $k$ or $k+1$ and is, respectively, a connected $k$-subset of $P_n^2$ or an interval of $k+1$ consecutive vertices. For fixed $r\ge3$, the formula is the restriction of a polynomial in $k$ of degree exactly $r-1$. Hence its $r$th backward difference vanishes identically and its $(r-1)$st backward difference is the constant $r-2$. On the topologically defined sequence, these identities hold for $k\ge r+2$ and $k\ge r+1$, respectively. We also obtain the conjectured closed forms for $k=4,5$, the complete face enumerator, and the associated $h$-polynomial and Stanley--Reisner Hilbert series.

math.CO↗

arXiv2026

Rational Weyl group elements of odd type D

Voloshyn introduced rational Weyl group elements in connection with rational normal forms on complex reductive groups and conjectured that their number in type $D_r$, for odd $r$, is $2^r-1$. We prove a stronger structural statement. For every odd integer $r\geq 5$, the rational elements of $W(D_r)$ are precisely the longest element $w_0$ and two explicitly described signed cyclic elements $c_I,d_I$ for each non-empty subset $I\subseteq\{1,\ldots,r-1\}$. Consequently, the rationality graph $\Gamma(D_r)$ is obtained by gluing two explicitly labelled subset-toggle graphs at $w_0$; it has $2^r-1$ vertices, and its only vertices of valency one are $c_{\{1\}}$ and $d_{\{1\}}$. The proof combines a two-level acyclic description of the root-poset graphs $\Gamma_{c_I}$ with a rigidity theorem for simple left multiplications of the signed cyclic family. A self-contained simply-laced reflection-preservation lemma and a terminal-layer argument provide the descent step, while every forbidden one-step move from the family is excluded by an explicit loop or two-cycle.

math.CO↗

arXiv2026

A Local Valuation Criterion for Quadratic-Permutation Interleaved Zadoff--Chu Sequences

Berggren and Popovi\'c introduced quadratic-permutation-polynomial interleaved Zadoff--Chu sequences and, from exhaustive data, conjectured that all normalized QPP-interleaved Zadoff--Chu sequences are inequivalent to ordinary Zadoff--Chu sequences precisely for prime-power lengths $N=p^n$ with $p>3$ and $n>1$. We give an exact local arithmetic criterion. For a normalized QPP $\pi_{a,b}(k)=ak^2+bk\pmod N$, the interleaved sequence is equivalent, under the standard five CAZAC-preserving operations, to a Zadoff--Chu sequence if and only if, for every prime power $p^\alpha\Vert N$, the valuation of $a$ satisfies \[ \nu_p(a)\ge \begin{cases} 0, & p=2,\ \alpha=1,\\ \alpha-1, & p=2,\ \alpha\ge2,\\ \alpha-1, & p=3,\\ \alpha, & p>3. \end{cases} \] The proof is based on a third finite-difference invariant of the lifted Zadoff--Chu phase, namely \[ \Delta^3\bigl((ak^2+bk+\varepsilon_N+2q)(ak^2+bk)\bigr) =12a(2ak+3a+b). \] As a consequence, the conjectured prime-power boundary is not correct: the exact non-vacuous condition for all nonzero normalized QPPs to be inequivalent to Zadoff--Chu sequences is that $N$ is odd, $9\nmid N$, and $p^2\mid N$ for at least one prime $p\ge5$. In particular, $N=75=3\cdot5^2$ is the smallest non-prime-power counterexample to the conjectured ``only if'' direction. A second corollary records the corresponding statement for irreducible QPPs.

math.NT↗

arXiv2026

Finite factorization is detected by undermonoids

Let $M$ be a cancellative commutative monoid and call a submonoid $S$ of $M$ an undermonoid if $\G(S)=\G(M)$ inside the Grothendieck group of $M$. Gotti and Li asked whether the finite factorization property is hereditary once it is known on all undermonoids: if every undermonoid of $M$ is a finite factorization monoid, must every submonoid of $M$ be a finite factorization monoid? We give an affirmative answer. Equivalently, for every cancellative commutative monoid $M$, the following two conditions coincide: every submonoid of $M$ is an FFM, and every undermonoid of $M$ is an FFM. The proof isolates a fixed length $\ell$ and an infinite set of length-$\ell$ factorizations of one element $b$. In the non-group case, a divisor-complement ideal $I=\{m\in M:m\nmid_M b\}$ enlarges the bad submonoid to a bad undermonoid while preserving the chosen length-$\ell$ factorizations. In the group case, a maximality argument over submonoids for which these factorizations survive is combined with a two-sided perturbation $S\mapsto S+\Nzero(2b+u)$. The key point is that the perturbation creates no new units and does not split any atom occurring in the fixed factorizations. This yields an undermonoid with infinitely many factorizations of $b$, contradicting the hypothesis.

math.GR↗

arXiv2026

A compensation theorem for the Sylow-integral invariant and counterexamples to an \texorpdfstring{$A_5$}{A5}-characterization conjecture

Let $\nu_p(G)$ be the number of Sylow $p$-subgroups of a finite group $G$, let $\sigma_p(G)$ be their common order, and set \[ \gamma(G)=\int_0^1\sum_{p\in\pi(G)}\nu_p(G)x^{\sigma_p(G)}\,dx =\sum_{p\in\pi(G)}\frac{\nu_p(G)}{\sigma_p(G)+1}. \] A recent conjectural extension of the simple-group theorem for this invariant asserted that a nonsolvable finite group has $\gamma(G)=9/2$ precisely when $G\cong A_5$. We disprove this assertion by a direct and verifiable construction. More generally, we prove an exact direct-product compensation formula for $A_5$ with an arbitrary nilpotent factor. The formula reduces the equality $\gamma(A_5\times N)=9/2$ to a finite Egyptian-fraction equation in the orders of the Sylow subgroups of $N$. Taking $N=\C_2\times\C_7\times\C_{11}\times\C_{13}\times\C_{17}\times\C_{19}\times\C_{29}\times\C_{71}\times\C_{83}$, the loss in the $2$-Sylow contribution is exactly compensated by the new normal Sylow subgroups. Consequently $G=A_5\times N$ is nonsolvable, is not isomorphic to $A_5$, has solvable radical $N$, and nevertheless satisfies $\gamma(G)=9/2$. Several further explicit compensation certificates are also recorded.

math.GR↗

arXiv2026

Weighted Uniform Endpoint Majorants for Integrals Involving Modified Bessel Functions

We give an affirmative full-range solution to Gaunt's 2019 Open Problem~2.10. The problem asks whether, for every $\nu>-1/2$ and $0<\gamma<1$, the reciprocal-power integral $\int_0^x e^{-\gamma t}I_\nu(t)t^{-\nu}\,\dd t$ is bounded by a constant multiple of $e^{-\gamma x}I_{\nu+1}(x)x^{-\nu}$, uniformly for all $x>0$. Earlier exponential-tilt estimates proved such endpoint majorants only under an additional smallness condition on $\gamma$. We prove the estimate throughout the natural range $0<\gamma<1$, with an explicit admissible constant. More generally, if $\mu>-1$, $q>-1$, $0<\gamma<1$, and $w(x)x^{-q}$ is nondecreasing on $(0,\infty)$, then for every $\theta\in(\gamma,1)$, $\int_0^x e^{-\gamma t}w(t)t^{-\mu}I_\mu(t)\,\dd t$ is controlled by an explicit multiple of $e^{-\gamma x}w(x)x^{-\mu}I_{\mu+1}(x)$. The case $w\equiv1$, $q=0$, and $\mu=\nu$ resolves Gaunt's problem. The weighted theorem also yields shifted-order and moment estimates, applies to approximate power weights and monotone regularly varying amplitudes, and provides two-sided estimates under a reversed comparison. We further analyze the sharp power-weighted quotient via endpoint expansions, a stationary equation, and parameter monotonicity.

math.CA↗

arXiv2026

A Determinant Congruence Conjectured by Sun

We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\in\Z$. If $n$ is composite, then \[ \det\big[(i^2+cij+dj^2)^{n-2}\big]_{0\leq i,j\leq n-1}\equiv 0\pmod {n^2} \] with no condition on $c$ and $d$. If $n=p$ is prime, the same congruence holds whenever the Legendre symbol $\leg{d}{p}$ is $-1$. For composite $n$, a polynomial determinant is divisible by two Vandermonde factors; after specialisation, their product already yields the required square divisor. For prime $n=p$, we estimate the rank of the matrix modulo $p$. The required rank defect follows from a coefficient cancellation obtained from the involution $t\mapsto d/t$ on $\Fp^\times$ and the condition $\leg{d}{p}=-1$.

math.NT↗

arXiv2026

Sun-type determinant and permanent congruences

Sun proposed a list of congruence and quadratic-residue conjectures for determinants and permanents over residue classes modulo a prime. This article gives a uniform treatment of Conjectures 4.6, 4.7, 4.8(ii), 4.9, 4.10(ii), 4.11 and 4.12 from Sun's list, while making explicit the overlap with two earlier contributions. Luo and Xia's Legendre-symbol formula for $D_p(b,1)$ already implies the non-vanishing assertion in Conjecture 4.6 when $p\equiv5\pmod {24}$; our determinant argument gives a root-quotient criterion for irreducible binary quadratic forms over $\Fp$ and also covers the remaining case $p\equiv19\pmod {24}$. For the Cauchy kernel $1/(x-y)$, we prove the derangement determinant and permanent congruences modulo $p^2$ and a polynomial fixed-point permanent congruence modulo $p$. For the Cayley kernel $(x+y)/(x-y)$, She, Sun and Xia's permanent identity supplies the structural input for the fixed-point permanent; combined with our Cauchy permanent congruence and Morley's congruence, it yields the congruence modulo $p^2$. Independent interpolation arguments give the signed fixed-point determinant congruences and the quadratic-residue assertion for the signed derangement determinant. Finally, a local expansion at the unique zero eigenvalue proves the half-size quadratic Cayley determinant divisibility by $p^2$, and by $p^3$ when $p\equiv7\pmod8$.

math.NT↗