arXiv subjects
Publications and source records attributed to Sergey Yurkevich.
We prove that the Nieuwland number of the Octahedron is $3\sqrt{2}/4$. Along the way, we provide an elementary proof that Rupert's problem for the Octahedron admits no solution in which the outer projection is a hexagon.
A three-dimensional convex body is said to have Rupert's property if its copy can be passed through a straight hole inside that body. In this work we construct a polyhedron which is provably not Rupert, thus we disprove a conjecture from 2017. We also find a polyhedron that is Rupert but not locally Rupert.
The combination of recent results due to Yu and Chen [Proc. AMS 150(4), 2020, 1749-1765] and to Bostan and Yurkevich [Proc. AMS 150(5), 2022, 2131-2136] shows that the 3-D Euclidean shape of the square Clifford torus is uniquely determined by its isoperimetric ratio. This solves part of the still open uniqueness problem of the Canham model for biomembranes. In this work we investigate the generalization of the aforementioned result to the case of a rectangular Clifford torus. Like the square case, we find closed-form formulas in terms of hypergeometric functions for the isoperimetric ratio of its stereographic projection to $\mathbb{R}^3$ and show that the corresponding function is strictly increasing. But unlike the square case, we show that the isoperimetric ratio does not uniquely determine the Euclidean shape of a rectangular Clifford torus.
In this article, we study systems of $n \geq 1$, not necessarily linear, discrete differential equations (DDEs) of order $k \geq 1$ with one catalytic variable. We provide a constructive and elementary proof of algebraicity of the solutions of such equations. This part of the present article can be seen as a generalization of the pioneering work by Bousquet-M\'elou and Jehanne (2006) who settled down the case $n=1$. Moreover, we obtain effective bounds for the algebraicity degrees of the solutions and provide an algorithm for computing annihilating polynomials of the algebraic series. Finally, we carry out a first analysis in the direction of effectivity for solving systems of DDEs in view of practical applications.
We provide a complete classification of the algebraicity of (generalized) hypergeometric functions with no restriction on the set of their parameters. Our characterization relies on the interlacing criteria of Christol (1987) and Beukers-Heckman (1989) for globally bounded and algebraic hypergeometric functions, however in a more general setting which allows arbitrary complex parameters with possibly integral differences. We also showcase the adapted criterion on a variety of different examples.
The $N$th power of a polynomial matrix of fixed size and degree can be computed by binary powering as fast as multiplying two polynomials of linear degree in~$N$. When Fast Fourier Transform (FFT) is available, the resulting complexity is \emph{softly linear} in~$N$, i.e.~linear in~$N$ with extra logarithmic factors. We show that it is possible to beat binary powering, by an algorithm whose complexity is \emph{purely linear} in~$N$, even in absence of FFT. The key result making this improvement possible is that the entries of the $N$th power of a polynomial matrix satisfy linear differential equations with polynomial coefficients whose orders and degrees are independent of~$N$. Similar algorithms are proposed for two related problems: computing the $N$th term of a C-finite sequence of polynomials, and modular exponentiation to the power $N$ for bivariate polynomials.
A constant term sequence is a sequence of rational numbers whose $n$-th term is the constant term of $P^n(\boldsymbol{x}) Q(\boldsymbol{x})$, where $P(\boldsymbol{x})$ and $Q(\boldsymbol{x})$ are multivariate Laurent polynomials. While the generating functions of such sequences are invariably diagonals of multivariate rational functions, and hence special period functions, it is a famous open question, raised by Don Zagier, to classify diagonals that are constant terms. In this paper, we provide such a classification in the case of sequences satisfying linear recurrences with constant coefficients. We also consider the case of hypergeometric sequences and, for a simple illustrative family of hypergeometric sequences, classify those that are constant terms.
We study systems of $n \geq 1$ discrete differential equations of order $k\geq1$ in one catalytic variable and provide a constructive and elementary proof of algebraicity of their solutions. This yields effective bounds and a systematic method for computing the minimal polynomials. Our approach is a generalization of the pioneering work by Bousquet-M\'elou and Jehanne (2006).
The technique of guessing can be very fruitful when dealing with sequences which arise in practice. This holds true especially when guessing is performed algorithmically and efficiently. One highly useful tool for this purpose is the package named $\texttt{gfun}$ in the software Maple. In this text we explore and explain some of $\texttt{gfun}$'s possibilities and illustrate them on two examples from recent mathematical research by the author and his collaborators.
A polyhedron $\textbf{P} \subset \mathbb{R}^3$ has Rupert's property if a hole can be cut into it, such that a copy of $\textbf{P}$ can pass through this hole. There are several works investigating this property for some specific polyhedra: for example, it is known that all 5 Platonic and 9 out of the 13 Archimedean solids admit Rupert's property. A commonly believed conjecture states that every convex polyhedron is Rupert. We prove that Rupert's problem is algorithmically decidable for polyhedra with algebraic coordinates. We also design a probabilistic algorithm which can efficiently prove that a given polyhedron is Rupert. Using this algorithm we not only confirm this property for the known Platonic and Archimedean solids, but also prove it for one of the remaining Archimedean polyhedra and many others. Moreover, we significantly improve on almost all known Nieuwland numbers and finally conjecture, based on statistical evidence, that the Rhombicosidodecahedron is in fact not Rupert.
We answer a question posed by Michael Aissen in 1979 about the $q$-analogue of a classical theorem of George P\'olya (1922) on the algebraicity of (generalized) diagonals of bivariate rational power series. In particular, we prove that the answer to Aissen's question, in which he considers $q$ as a variable, is negative in general. Moreover, we show that when $q$ is a complex number, the answer is positive if and only if $q$ is a root of unity.
We provide a short and elementary proof of the main technical result of the recent article "On the uniqueness of Clifford torus with prescribed isoperimetric ratio" by Thomas Yu and Jingmin Chen. The key of the new proof is an explicit expression of the central function (${\sf Iso}$, to be proved bijective) as a quotient of Gaussian hypergeometric functions.
In 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial $N!$ in arithmetic complexity quasi-linear in $\sqrt{N}$. In 1988, the Chudnovsky brothers generalized Strassen's algorithm to the computation of the $N$-th term of any holonomic sequence in essentially the same arithmetic complexity. We design $q$-analogues of these algorithms. We first extend Strassen's algorithm to the computation of the $q$-factorial of $N$, then Chudnovskys' algorithm to the computation of the $N$-th term of any $q$-holonomic sequence. Both algorithms work in arithmetic complexity quasi-linear in $\sqrt{N}$; surprisingly, they are simpler than their analogues in the holonomic case. We provide a detailed cost analysis, in both arithmetic and bit complexity models. Moreover, we describe various algorithmic consequences, including the acceleration of polynomial and rational solving of linear $q$-differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost.
There are many viewpoints on algebraic power series, ranging from the abstract ring-theoretic notion of Henselization to the very explicit perspective as diagonals of certain rational functions. To be more explicit on the latter, Denef and Lipshitz proved in 1987 that any algebraic power series in $n$ variables can be written as a diagonal of a rational power series in one variable more. Their proof uses a lot of involved theory and machinery which remains hidden to the reader in the original article. In the present work we shall take a glimpse on these tools by motivating while defining them and reproving most of their interesting parts. Moreover, in the last section we provide a new significant improvement on the Artin-Mazur lemma, proving the existence of a 2-dimensional code of algebraic power series.
The First Fundamental Theorem of Invariant Theory describes a minimal generating set of the invariant polynomial ring under the action of some group $G$. In this note we give an elementary and direct proof for the $\operatorname{GL}_2(K)$ and $\operatorname{SL}_2(K)$ for any infinite field $K$. Our proof can be generalized to $\operatorname{GL}_m(K)$ and $\operatorname{SL}_m(K)$ for $m>2$. Moreover, we present a family of counter-examples to the statements of the First Fundamental Theorems for all finite fields and $m=2$.
We prove that the diagonal of any finite product of algebraic functions of the form \begin{align*} {(1-x_1-\dots-x_n)^R}, \qquad R\in\mathbb{Q}, \end{align*} is a generalized hypergeometric function, and we provide an explicit description of its parameters. The particular case $(1-x-y)^R/(1-x-y-z)$ corresponds to the main identity of Abdelaziz, Koutschan and Maillard in [AKM2020, {\S}3.2]. Our result is useful in both directions: on the one hand it shows that Christol's conjecture holds true for a large class of hypergeometric functions, on the other hand it allows for a very explicit and general viewpoint on the diagonals of algebraic functions of the type above. Finally, in contrast to [AKM2020], our proof is completely elementary and does not require any algorithmic help.