arXiv subjects

Arthemy V. Kiselev

Publications and source records attributed to Arthemy V. Kiselev.

At least 19 recordsLinked to original sources

arXiv2026

New integer sequence OEIS A392714 counts Wronskians: fast evaluation via late-growing permutations

The alternating composition of $N = 2p$ weighted differential operators $w_j(x)\cdot\partial_x^{\,p}$ of strict order $p$ on the line $\mathbb{R} \ni x$ is again an operator of order $p$; its coefficient is the universal constant $c(p)$ times the Wronskian of the weights $w_1,\ldots,w_N$. Lie brackets of vector fields fix $c(p=1)=1$; we want to find $c(p \geqslant 2)$: e.g., $c(2) = 2$ or $c(3) = 90$. Direct symbolic expansion (over $|S_{2p}| =(2p)!$ permutations) fails for $p \geqslant 4$. Taking the monomials $w_j = x^{j-1}$ reduces the summation to the much smaller set $Φ_p \subseteq S_{2p-1} \subsetneq S_{2p}$ of late-growing permutations. Expressing $c(p)$ as a signed sum of products of falling factorials, we implement and speed up the algorithm that gains all the integer values up to $c(18) = 4.881\ldots \cdot 10^{462}$. The resulting sequence is new, now registered as OEIS A392714; its (sub)leading-order growth rate is $\log c(p) \simeq 2p^2\log p -b p^2 + \overline{o}(p^2)$ for $p\gg 1$, with $b\geqslant 2.6744$.

math.NT↗

arXiv2026

Iterate Wronskians over $\mathbb{R}^d$ as $N$-ary brackets on $\mathbb{R}[x^1,\ldots,x^d]$: the $N$-bonacci numbers bound the highest total degrees

For the algebra $\mathbb{R}[x^1,\ldots,x^d]$ of polynomials in $d\geqslant 1$ variables, regard the complete generalised Wronskian $W_d^k$ of differential order $k\geqslant 1$ over $\mathbb{R}^d$ as the $N=\tbinom{d+k}{d}$-ary Lie bracket. Take an $N$-tuple of polynomials, calculate their Wronskian, and keep re-using the newly-created polynomials to produce more of them. The problem is: how fast do their maximal total degrees grow with the number $n$ of iterations of the bracket? Here enter the $N$-bonacci numbers defined by the recurrence $F^{(N)}_n=F^{(N)}_{n-1}+\cdots+F^{(N)}_{n-N}\in \mathbb{N}$. We prove that for any choice of the initial arguments, the sequence of highest total degrees ${\mathsf{D}}^{(N)}_n \geqslant 0$ grows (if at all) asymptotically no faster than the $n$th $N$-bonacci number: $\lim_{n\to+\infty} ({\mathsf{D}}^{(N)}_n/F^{(N)}_n )<\infty$. We show that for $d=1$ and $k$ odd, the highest polynomial degrees do attain the $N$-bonacci bound. Keywords: Differential polynomial, $N$-ary Lie bracket, multivariate Wronskian determinant, Fibonacci numbers, $N$-bonacci numbers, asymptotic growth rate, growth of polynomial degrees, Skolem--Pisot problem.

math.RA↗

arXiv2026

The alternating compositions of weighted differential operators yield the weights' Wronskian with which constant?

The alternated composition of $N=2p$ differential operators $w_j(x)\,\partial_x^p$ of strict order $p$ on the line $\mathbb{R}\ni x$ is again a differential operator of strict order $p$; its coefficient is the constant $\mathrm{const}(p)$, depending only on the arity $N$, times the Wronskian determinant of the originally taken coefficients $w_1,\dots,w_N$. The case $p=1$ of the Lie bracket for two vector fields fixes $\mathrm{const}(1)=1$, and $\mathrm{const}(2)=2$ is found easily by hand; $\mathrm{const}(3)=90$ can still be obtained symbolically. The problem is to determine $\mathrm{const}(p\geqslant4)$. We compute $\mathrm{const}(p)$ exactly for all $p\leqslant14$ -- a 241-digit integer at $p=14$ -- and record the resulting integer sequence as OEIS A392714. We prove that $v_p(\mathrm{const}(p))\geqslant p-1$ for every prime $p$, matching the exact equality observed numerically throughout our range, and conjecture that this equality holds in general. We show that $\log\mathrm{const}(p)$ grows like $αp^2\log p$, with the leading coefficient close to $2$, nearly saturating the bound $p^2\log p\,(1+O(1/\log p))\leqslant\log\mathrm{const}(p)\leqslant2p^2\log p\,(1+O(1/\log p))$ obtained by O. Zaboronski (private communication). A naturally arising reduced constant is found to decay to zero super-exponentially rather than grow, a direct consequence of this near-saturation.

math.CO↗

arXiv2026

Explicit class of finite-dimensional polynomial algebras with Wronskians over $\mathbb{R}^d$ as $N$-ary Lie brackets: beyond $\mathfrak{sl}(2)$

Lie algebra $\mathfrak{sl}(2)$ can be realised by vector fields on $\mathbb{R}^1\ni x$ with polynomial coefficients $1$, $-2x$, $-x^2$; their Wronskian determinants yield the Lie bracket. Likewise, the monomials $1$, $\ldots$, $x^k/k!$, $\ldots$, $x^N/N!$ span finite-dimensional strong homotopy (SH) Lie algebras with the Wronskians $\mathbf{1} \wedge \partial_x \wedge \ldots \wedge \partial_x^{N-1}$ as the $N$-ary brackets. Over dimension $d=2$ with $\mathbb{R}^2\ni(x,y)$ and for the complete generalised Wronskian $W_{d=2}^{k=1}=\mathbf{1}\wedge \partial_x \wedge \partial_y$ of differential order $k=1$ as the ternary bracket, the finite-dimensional polynomial SH-Lie algebras are spanned by $\langle 1$, $x$, $y$, $p\rangle$ with $p\in\{x^2$, $xy$, $y^2\}$. We explicitly describe all finite-dimensional polynomial SH-Lie algebras $\Bbbk_k[{\boldsymbol{x}}]\subseteq \mathcal{A} \subseteq \Bbbk[x^1,\ldots,x^d]$ (over $\Bbbk=\mathbb{R}$ or $\mathbb{C}$) with the complete generalised Wronskians $W_{d\geqslant 1}^{k\geqslant 1}$ of order $k$ as $N$-ary brackets: $N=\binom{d+k}{d}$. We obtain a factorisation formula for the generalised Vandermonde determinants which show up in the structure constants of the polynomial algebras $\mathcal{A}$.

math.RA↗

arXiv2025

Kontsevich graphs act on Nambu-Poisson brackets, VI. Open problems

Kontsevich's graphs from deformation quantisation allow encoding multi-vectors whose coefficients are differential-polynomial in components of Poisson brackets on finite-dimensional affine manifolds. The calculus of Kontsevich graphs can be made dimension-specific for the class of Nambu--Poisson brackets given by Jacobian determinants. Using the Kontsevich--Nambu micro-graphs in dimensions $d\geqslant 2$, we explore the open problem of (non)triviality for Kontsevich's tetrahedral graph cocycle action on the space of Nambu--Poisson brackets. We detect a conjecturally infinite new set of differential-polynomial identities for Jacobian determinants of arbitrary sizes $d\times d$.

math.CO↗

arXiv2025

Jacobi identities for Wronskian determinants over multidimension

The generalised Wronskian of differential order $k\geqslant 1$ for $N$ functions $f_1$, $\ldots$, $f_N$ in $d\geqslant 1$ independent variables $x^1$, $\ldots$, $x^d$ is the determinant of the matrix with these functions' derivatives $\partial^{|σ_i|} f_j / \partial (x^1)^{σ_i^1}\cdots \partial (x^d)^{σ_i^d}$ (of orders $0 \leqslant |σ_i| \leqslant k$), where the multi-indices $σ_i$ mark (all or part of) fibre variables $u_{σ_i}$ in the $k$th jet space $J^k\bigl(\mathbb{R}^d\to\mathbb{R}\bigr)$. We prove that these (in)complete Wronskians -- provided that their lowest-order parts are complete at differential orders $\ell\leqslant 1$ -- over the $d$-dimensional base satisfy the table of bi-linear, Jacobi-type identities for Schlessinger--Stasheff's strongly homotopy Lie algebras.

math.RA↗

arXiv2025

Wronskians as $N$-ary brackets in finite-dimensional analogues of $sl(2)$

The Wronskian determinants (for coefficients of higher-order differential operators on the affine real line or circle) satisfy the table of Jacobi-type quadratic identities for strong homotopy Lie algebras -- i.e. for a particular case of $L_\infty$-deformations -- for the Lie algebra of vector fields on that one-dimensional affine manifold. We show that the standard realisation of $\mathfrak{sl}(2)$ by quadratic-coefficient vector fields is the bottom structure in a sequence of finite-dimensional polynomial algebras $\Bbbk_N[x]$ with the Wronskians as $N$-ary brackets; the structure constants are calculated explicitly. Key words: Wronskian determinant, $N$-ary bracket, $L_\infty$-\/algebra, strong homotopy Lie algebra, $sl(2)$, Witt algebra, Vandermonde determinant.

math.RA↗

arXiv2025

Kontsevich graphs act on Nambu--Poisson brackets, IV. When the invisible becomes crucial

Kontsevich's graphs allow encoding multi-vectors whose coefficients are differential-polynomial in the coefficients of a given Poisson bracket on an affine real manifold. Encoding formulas by directed graphs adapts to the class of Nambu-determinant Poisson brackets, yet the graph topology becomes dimension-specific. To inspect whether a given Kontsevich graph cocycle $γ$ acts (non)trivially -- in the second Poisson cohomology -- on the space of Nambu brackets, taking a vector field solution $\smash{\vec{X}^γ_d}$ from dimension $d$ does not work in $d+1$. For $2 \leqslant d \leqslant 4$, the action of tetrahedron $γ_3$ on Nambu brackets is known to be a Poisson coboundary, $\dot{P} = [[ P,\smash{\vec{X}^{γ_3}_d} (P)]]$. We explore which minimal (sub)sets of graphs, encoding (non)vanishing objects over $\mathbb{R}^d_{\text{aff}}$, generate the topological data that suffice for a solution $\smash{\vec{X}^{γ_3}_{d+1}}$ to appear. We detect that there can be no solution in higher dimension without invisible graphs that vanish as formulas in $d=3$, but whose descendants do not all vanish over $d=4$.

math.CO↗

arXiv2025

Kontsevich graphs act on Nambu--Poisson brackets, V. Implementation

In this series of papers, we established that $Q^{γ_3}_{d=4}(P)$ is a coboundary in 4D (paper II arXiv:2409.12555), and we presented a series of experimental results about the (non)trivialisation of Kontsevich graph flows of Nambu--Poisson brackets on $\mathbb{R}^d$ (paper IV). This immediate sequel V. to I.--IV. is a guide to working with the package $\textsf{gcaops}$ (https://github.com/rburing/gcaops) ($\textbf{G}$raph $\textbf{C}$omplex $\textbf{A}$ction $\textbf{O}$n $\textbf{P}$oisson $\textbf{S}$tructures) for $\textsf{SageMath}$ by Buring (2022). Specifically, we shall explain the script used in paper II (arXiv:2409.12555) and the use of it.

math.CO↗

arXiv2024

Kontsevich graphs act on Nambu--Poisson brackets, II. The tetrahedral flow is a coboundary in 4D

Kontsevich constructed a map from suitable cocycles in the graph complex to infinitesimal deformations of Poisson bi-vector fields. Under the deformations, the bi-vector fields remain Poisson. We ask, are these deformations trivial, meaning, do they amount to a change of coordinates along a vector field? We examine this question for the tetrahedron, the smallest nontrivial suitable graph cocycle in the Kontsevich graph complex, and for the class of Nambu--Poisson brackets on $\mathbb{R}^d$. Within Kontsevich's graph calculus, we use dimension-specific micro-graphs, in which each vertex represents an ingredient of the Nambu--Poisson bracket. For the tetrahedron, Kontsevich knew that the deformation is trivial for $d = 2$ (1996). In 2020, Buring and the third author found that the deformation is trivial for $d = 3$. Building on these discoveries, we now establish that the deformation is trivial for $d = 4$.

math.QA↗

arXiv2024

Kontsevich's star-product up to order 7 for affine Poisson brackets: where are the Riemann zeta values?

The Kontsevich star-product admits a well-defined restriction to the class of affine -- in particular, linear -- Poisson brackets; its graph expansion consists only of Kontsevich's graphs with in-degree $\leqslant 1$ for aerial vertices. We obtain the formula $\star_{\text{aff}}\text{ mod }\bar{o}(\hbar^7)$ with harmonic propagators for the graph weights (over $n\leqslant 7$ aerial vertices); we verify that all these weights satisfy the cyclic weight relations by Shoikhet--Felder--Willwacher, that they match the computations using the $\textsf{kontsevint}$ software by Panzer, and the resulting affine star-product is associative modulo $\bar{o}(\hbar^7)$. We discover that the Riemann zeta value $ζ(3)^2/π^6$, which enters the harmonic graph weights (up to rationals), actually disappears from the analytic formula of $\star_{\text{aff}}\text{ mod }\bar{o}(\hbar^7)$ \textit{because} all the $\mathbb{Q}$-linear combinations of Kontsevich graphs near $ζ(3)^2/π^6$ represent differential consequences of the Jacobi identity for the affine Poisson bracket, hence their contribution vanishes. We thus derive a ready-to-use shorter formula $\star_{\text{aff}}^{\text{red}}$ mod~$\bar{o}(\hbar^7)$ with only rational coefficients.

math.QA↗

arXiv2024

Kontsevich graphs act on Nambu-Poisson brackets, I. New identities for Jacobian determinants

Nambu-determinant brackets on $R^d\ni x=(x^1,...,x^d)$, $\{f,g\}_d(x)=ρ(x) \det(\partial(f,g,a_1,...,a_{d-2})/\partial(x^1,...,x^d))$, with $a_i\in C^\infty(R^d)$ and $ρ\partial_x\in\mathfrak{X}^d(R^d)$, are a class of Poisson structures with (non)linear coefficients, e.g., polynomials of arbitrarily high degree. With good cocycles in the graph complex, Kontsevich associated universal -- for all Poisson bivectors $P$ on affine $R^d_{aff}$ -- elements $\dot{P}=Q^γ(P)\in H^2_{P}(R^d_{aff})$ in the Lichnerowicz-Poisson second cohomology groups; we note that known graph cocycles $γ$ preserve the Nambu-Poisson class $\{P(ρ,a)\}$, and we express, directly from $γ$, the evolution $\dotρ$,$\dot{a}$ that induces $\dot{P}$. Over all $d\geq2$ at once, there is no universal mechanism for the bivector cocycles $Q^γ_d$ to be trivial, $Q^γ_d=[\![P,\vec{X}^γ_d(P)]\!]$, w.r.t. vector fields defined uniformly for all dimensions $d$ by the same graph formula. While over $R^2$, the graph flows $\dot{P} = Q^{γ_i}_{2D}(P(ρ))$ for $γ\in\{γ_3,γ_5,γ_7,...\}$ are trivialized by vector fields $\vec{X}^{γ_i}_{2D}=(dx\wedge dy)^{-1}d_{dR}(Ham^{γ_i}(P))$ of peculiar shape, we detect that in $d\geq3$, the 1-vectors from 2D, now with $P(ρ,a_1,...,a_{d-2})$ inside, do not solve the problems $Q^{γ_i}_{d\geq3}=[\![P,{\vec{X}^{γ_i}_{d\geq3}}(P(ρ,a))]\!]$, yet they do yield good Ansatz where we find solutions $\vec{X}^{γ_i}_{d=3,4}(P(ρ,a))$. In the study of the step $d\mapsto d+1$, by adapting the Kontsevich graph calculus to the Nambu-Poisson class of brackets, we discover more identities for the Jacobian determinants within $P(ρ,a)$, i.e. for multivector-valued $GL(d)$-invariants on $R^d_{aff}$.

math.QA↗

arXiv2023

Associativity certificates for Kontsevich's star-product $\star$ mod $\bar{o}(\hbar^k)$: $k\leqslant 6$ unlike $k\geqslant7$

The formula $\star$ mod $\bar{o}(\hbar^k)$ of Kontsevich's star-product with harmonic propagators was known in full at $\hbar^{k\leqslant 6}$ since 2018 for generic Poisson brackets, and since 2022 also at $k=7$ for affine brackets. We discover that the mechanism of associativity for the star-product up to $\bar{o}(\hbar^6)$ is different from the mechanism at order $7$ for both the full star-product and the affine star-product. Namely, at lower orders the needed consequences of the Jacobi identity are immediately obtained from the associator mod $\bar{o}(\hbar^6)$, whereas at order $\hbar^7$ and higher, some of the necessary differential consequences are reached from the Kontsevich graphs in the associator in strictly more than one step.

math.QA↗

arXiv2023

The tower of Kontsevich deformations for Nambu-Poisson structures on $\mathbb{R}^{d}$: dimension-specific micro-graph calculus

In Kontsevich's graph calculus, internal vertices of directed graphs are inhabited by multi-vectors, e.g., Poisson bi-vectors; the Nambu-determinant Poisson brackets are differential-polynomial in the Casimir(s) and density $\varrho$ times Levi-Civita symbol. We resolve the old vertices into subgraphs such that every new internal vertex contains one Casimir or one Levi-Civita symbol${}\times\varrho$. Using this micro-graph calculus, we show that Kontsevich's tetrahedral $γ_3$-flow on the space of Nambu-determinant Poisson brackets over $\mathbb{R}^3$ is a Poisson coboundary: we realize the trivializing vector field $\smash{\vec{X}}$ over $\smash{\mathbb{R}^3}$ using micro-graphs. This $\smash{\vec{X}}$ projects to the known trivializing vector field for the $γ_3$-flow over $\smash{\mathbb{R}^2}$.

math.CO↗

arXiv2022

The hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets

Kontsevich's graph flows are -- universally for all finite-dimensional affine Poisson manifolds -- infinitesimal symmetries of the spaces of Poisson brackets. We show that the previously known tetrahedral flow and the recently obtained pentagon-wheel flow preserve the class of Nambu-determinant Poisson bi-vectors $P=[\![ \varrho(\boldsymbol{x})\,\partial_x\wedge\partial_y\wedge\partial_z,a]\!]$ on $\mathbb{R}^3\ni\boldsymbol{x}=(x,y,z)$ and $P=[\![ [\![\varrho(\boldsymbol{y})\,\partial_{x^1}\wedge\ldots\wedge\partial_{x^4},a_1]\!],a_2]\!]$ on $\mathbb{R}^4\ni\boldsymbol{y}$, including the general case $\varrho \not\equiv 1$. We detect that the Poisson bracket evolution $\dot{P} = Q_γ(P^{\otimes^{\# Vert(γ)}})$ is trivial in the second Poisson cohomology, $Q_γ= [\![ P, \vec{X}([\varrho],[a]) ]\!]$, for the Nambu-determinant bi-vectors $P(\varrho,[a])$ on $\mathbb{R}^3$. For the global Casimirs $\mathbf{a} = (a_1,\ldots,a_{d-2})$ and inverse density $\varrho$ on $\mathbb{R}^d$, we analyse the combinatorics of their evolution induced by the Kontsevich graph flows, namely $\dot{\varrho} = \dot{\varrho}([\varrho], [\mathbf{a}])$ and $\dot{\mathbf{a}} = \dot{\mathbf{a}}([\varrho],[\mathbf{a}])$ with differential-polynomial right-hand sides. Besides the anticipated collapse of these formulas by using the Civita symbols (three for the tetrahedron $γ_3$ and five for the pentagon-wheel graph cocycle $γ_5$), as dictated by the behaviour $\varrho(\mathbf{x}') = \varrho(\mathbf{x}) \cdot \det \| \partial \mathbf{x}' / \partial \mathbf{x} \|$ of the inverse density $\varrho$ under reparametrizations $\mathbf{x} \rightleftarrows \mathbf{x}'$, we discover another, so far hidden discrete symmetry in the construction of these evolution equations.

math.SG↗

arXiv2020

The Kontsevich graph orientation morphism revisited

The orientation morphism $Or(\cdot)(P)\colon γ\mapsto\dot{P}$ associates differential-polynomial flows $\dot{P}=Q(P)$ on spaces of bi-vectors $P$ on finite-dimensional affine manifolds $N^d$ with (sums of) finite unoriented graphs $γ$ with ordered sets of edges and without multiple edges and one-cycles. It is known that $d$-cocycles $\boldsymbolγ\in\ker d$ with respect to the vertex-expanding differential $d=[{\bullet}\!\!{-}\!{-}\!\!{\bullet},\cdot]$ are mapped by $Or$ to Poisson cocycles $Q(P)\in\ker\,[\![ P,{\cdot}]\!]$, that is, to infinitesimal symmetries of Poisson bi-vectors $P$. The formula of orientation morphism $Or$ was expressed in terms of the edge orderings as well as parity-odd and parity-even derivations on the odd cotangent bundle $ΠT^* N^d$ over any $d$-dimensional affine real Poisson manifold $N^d$. We express this formula in terms of (un)oriented graphs themselves, i.e. without explicit reference to supermathematics on $ΠT^* N^d$.

math.CO↗

arXiv2019

Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms

The graph complex acts on the spaces of Poisson bi-vectors $P$ by infinitesimal symmetries. We prove that whenever a Poisson structure is homogeneous, i.e. $P = L_{\vec{V}}(P)$ w.r.t. the Lie derivative along some vector field $\vec{V}$, but not quadratic (the coefficients of $P$ are not degree-two homogeneous polynomials), and whenever its velocity bi-vector $\dot{P}=Q(P)$, also homogeneous w.r.t. $\vec{V}$ by $L_{\vec{V}}(Q)=n\cdot Q$ whenever $Q(P)= Or(γ)(P^{\otimes^n})$ is obtained using the orientation morphism $Or$ from a graph cocycle $γ$ on $n$ vertices and $2n-2$ edges in each term, then the $1$-vector $\vec{X}=Or(γ)(\vec{V}\otimes P^{\otimes^{n-1}})$ is a Poisson cocycle. Its construction is uniform for all Poisson bi-vectors $P$ satisfying the above assumptions, on all finite-dimensional affine manifolds $M$. Still, if the bi-vector $Q\not\equiv 0$ is exact in the respective Poisson cohomology, so there exists a vector field $\vec{Y}$ such that $Q(P)=[\![\vec{Y},P]\!]$, then the universal cocycle $\vec{X}$ does not belong to the coset of $\vec{Y}$ mod $\ker[\![P,\cdot]\!]$. We illustrate the construction using two examples of cubic-coefficient Poisson brackets associated with the $R$-matrices for the Lie algebra $\mathfrak{gl}(2)$. Keywords: Graph complex, Poisson bracket, deformation cohomology, affine manifold, tetrahedral flow, diffeomorphism.

math.SG↗

arXiv2019

Open problems in the Kontsevich graph construction of Poisson bracket symmetries

Poisson brackets admit infinitesimal symmetries which are encoded using oriented graphs; this construction is due to Kontsevich (1996). We formulate several open problems about combinatorial and topological properties of the graphs involved, about integrability and analytic properties of such symmetry flows (in particular, for known classes of Poisson brackets), and about cohomological, differential geometric, and quantum aspects of the theory. Keywords: Poisson bracket, Jacobi identity, infinitesimal symmetry, affine manifold, Poisson cohomology, graph complex.

math-ph↗