arXiv subjects

David Kumallagov

Publications and source records attributed to David Kumallagov.

8 recordsLinked to original sources

arXiv2026

Uniform Rost nilpotence and birational motives

For a field extension $E/k$ and a Chow motive $M$, let $I_E(M)=\ker\bigl(\operatorname{End}_k(M)\longrightarrow \operatorname{End}_E(M_E)\bigr)$ be the base-change ideal. We establish explicit nilpotence bounds for these ideals in several geometric settings. We also prove effective generic descent: if $M$ is a summand of $h(X)(a)$ and $M_{k(X)}$ has uniform Rost exponent $s$, then $M$ has exponent $s(\dim X+1)$. This yields the integral exponent $2n-1$ for twisted Milnor hyperplane sections of dimension $2n-2$. In characteristic zero, a weighted local-block refinement of the multilinear Rost filtration improves the uniform bounds of Gille and upgrades the elementwise estimates of Rosenschon--Sawant to uniform bounds at the ideal level. Using the Kahn--Sujatha description of pure birational motives, we obtain quantitative lifting results from birational to ordinary Chow motives, with applications to threefolds, including varieties birational to toric models and varieties admitting a decomposition of the diagonal supported on a surface. Finally, we extend the Kok--Zhou detector from individual correspondences to entire base-change ideals, obtaining an explicit ideal nilpotence bound from uniform annihilation of the critical refined unramified cohomology groups.

math.AG↗

arXiv2026

Picard groups and composition nilpotence for finite cellular isotropic spectra

Let $k=k_0(t_1,t_2,\ldots)$ be a flexible field of characteristic different from $2$, let $\mathbb X$ be the mod-$2$ isotropic sphere, and set $E=\mathbb X \wedge MBP$. We prove $$Pic\bigl(SH(k/k)^c_{cell}\bigr)\cong\mathbb Z^2; $$ thus every tensor-invertible finite cellular isotropic spectrum is a unique bigraded suspension of $\mathbb X$. More generally, $E_{**}$ is conservative on finite cellular objects, and concentration on one diagonal forces a finite direct sum of suspended isotropic spheres. A bounded diagonal weight structure recovers the exact weights and minimal-complex terms from $E_{**}$. For every nonzero finite cellular $M$, the kernel of $$End(M)\longrightarrow End(E\wedge_{\mathbb X}M)$$ is a composition-nilpotent ideal, with exponent at most $d(M)(2L(M)-1)$, where $L(M)$ is the diagonal width and $d(M)$ the maximal number of distinct Tate degrees on one diagonal. This yields detection of composition nilpotence and canonical Fitting decompositions.

math.AG↗

arXiv2026

Frobenius orbit codes and finite-field Fourier spectra: exact distance and coset distributions

Let $q$ be a prime power, let $m\ge1$, put $Q=q^m$, and let a permutation $π$ of a finite index set $I$ satisfy $π^m=1$. We study the $\F q$-linear code \[ \mathcal C(π)=\{x\in\Lfield^I:x_{π(i)}=x_i^q\}, \] which includes the Fourier spectra of $\K$-valued functions on finite abelian groups split by $\Lfield$. Our structural starting point is a $\F q$-linear Hamming isometry taking each cycle of $π$ of length $\ell$ to the subfield repetition code $\{(b,\ldots,b):b\in\F{q^\ell}\}$. From this normal form we obtain exact symbol weight and received-word distance enumerators, all list sizes, and the covering radius. An occupancy generating function gives the complete coset-leader distribution, the exact mean distance, the probability of a unique nearest word, and the number of deep holes. For the full cyclic family of length $q^m-1$, the deficit of a uniformly random ambient word from the covering radius is asymptotically Poisson with mean $(m-1)/2$, with an additional $1$ when $m$ is even; a central limit theorem follows. We also determine the deep-hole probability and the second-order gap to the sphere-covering bound. Classical Fourier--Galois descent and cyclotomic orbit enumeration enter only to identify the Fourier specialization.

math.AC↗

arXiv2026

Affine monodromy and exact value distributions over finite fields

We study finite field value distributions through the fixed-point statistics of monodromy groups. For a regular degree-$n$ cover, omitted values are controlled by derangements. Thus natural symmetric monodromy gives support density $1-D_n/n!\to 1-e^{-1}$, while the Cameron--Cohen bound gives the universal ceiling $1-1/n$, attained by sharply $2$-transitive affine monodromy. We give an explicit polynomial realization of this optimal mechanism. For $N=p^e$ and $h\mid N-1$, set \[ Λ_{N,h}(U)=U\bigl(U^{(N-1)/h}-1\bigr)^h . \] Its geometric Galois closure is rational, \[ U=z^h, \qquad T=(z^N-z)^h, \] and its geometric monodromy is the affine group $(\F_N,+)\rtimes H_h$. For every extension $\F_Q/\F_p$ we compute the complete fibre enumerator of $Λ_{N,h}$ exactly, including the nonregular cases. In the full affine case $h=N-1$, the polynomial \[ U(U-1)^{N-1} \] attains the Wan--Shiue--Chen upper bound for non-permutation polynomials over every finite field containing $\F_N$; over arbitrary extensions we compute the exact defect from that bound.

math.NT↗

arXiv2026

Dimension filtrations in birational localisation

Let $S_b$ be the class of birational morphisms between smooth varieties over a field $F$, and let $L_n=S_b^{-1}d_{\leq n}\Sm(F)$. Kahn and Sujatha asked whether the natural functor $L_n\to S_b^{-1}\Sm(F)$ is fully faithful. We prove that it is fully faithful exactly for $n=0$. More strongly, for every $n\geq1$ and every $N\geq n+1$, the transition functor $L_n\to L_N$ has an infinite fibre on an endomorphism set. The proof identifies a sharp dimension threshold: if $\dim X+r\leq n$, then $X\times\mathbb A^r\to X$ is invertible in $L_n$ precisely when $\dim X+r\leq n-1$. We also give proper and projective analogues.

math.AG↗

arXiv2026

A coproduct obstruction for derived unramified cohomology

Let $k$ be a perfect field of exponential characteristic $p$, and let $R$ be a commutative $\Z[1/p]$-algebra. We prove that the first derived unramified functor \[ F\longmapsto R^1_{\mathrm{nr},R}F \] from homotopy invariant Nisnevich sheaves with transfers of $R$-modules to birational sheaves commutes with arbitrary small direct sums. This gives a positive answer, after inverting the exponential characteristic, to a question of Kahn and Sujatha; on smooth projective varieties no inversion is needed. We also describe an obstruction to this for the functor $R^2_{\mathrm{nr},R}$ in categorical terms, which includes the familiar Griffiths group obstruction. As applications of the motivic nature of the functors $R^q_{\mathrm{nr}}$, we prove torsion-order bounds and a correspondence-detection statement for surfaces.

math.AG↗

arXiv2026

Exact output statistics of Icart's encoding in the exceptional $j=0$ case

Icart's encoding is a classical deterministic map from finite fields to elliptic curves and a basic ingredient in early hash-to-curve constructions. We determine the exact one-output distribution of this map in the exceptional $j=0$ case. More precisely, for \[ E_{0,b}:Y^2=X^3+b,\ q\equiv2\pmod3, \] we compute the complete fibre distribution of $f_{0,b}:\mathbb F_q\to E_{0,b}(\mathbb F_q)$. This gives closed formulae for the image size, total variation distance from uniform, collision probability, power sums, entropy measures and basic batch statistics. We also derive the exact second moment of all nontrivial character sums of the output distribution. Via the Weil pairing, this becomes an exact energy formula for pairing-character tests on the supersingular $j=0$ family whose odd prime order subgroups have embedding degree two.

math.NT↗

arXiv2026

An extremal problem for completely unclustered Burrows-Wheeler images

The Burrows--Wheeler transform is usually viewed as a clustering transform: it tends to group equal letters into long runs. We study the opposite extremal regime, where the BWT output is completely unclustered, that is, has as many equal-letter runs as positions. Known results imply, on the one hand, that the number of runs in the BWT of a Lyndon word can increase by at most a factor of two, and, on the other hand, that over every alphabet of size at least three completely unclustered BWT images exist in every length. This leads to the extremal problem lying between these two facts. For $k\ge3$, let $U_k(n)$ be the minimum cyclic run number of a primitive necklace of length $n$ whose BWT has $n$ runs. We prove the universal lower bound $U_k(n)\ge\lceil n/2\rceil$, reduce the sharpness problem for one-cycle BWT images $L$ to the Hamming identity \[ \cruns(\BWT^{-1}(L))=\dH(L,\sort(L)), \] and develop a natural multiset-of-necklaces relaxation with an explicit constant-cycle correction. We compute the small values, including the exceptional value $U_k(6)=4$, prove a parity obstruction for the Parikh vectors of sharp examples, and determine the multiset relaxation exactly. Finally, for every prime $p\equiv5\pmod8$ for which $2$ is a primitive root modulo $p$, we prove sharpness in the adjacent lengths $p-1$ and $p$. Under the corresponding Artin-type infinitude hypothesis, this gives infinitely many adjacent sharp pairs.

math.CO↗