Journal of Symbolic Computation最新文献

We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations are algebraic (ordinary or partial) differential equations (ADEs). The general purpose is to find ADEs whose solutions contain specified rational expressions of solutions to given ADEs. For univariate D-algebraic functions, we show how to derive an ADE of smallest possible order. In the multivariate case, we introduce a general algorithm for these computations and derive conclusions on the order bound of the resulting algebraic PDE. Using our accompanying Maple software, we discuss applications in physics, statistics, and symbolic integration.

We illustrate the power of symbolic computation and experimental mathematics by investigating maximal seating arrangements, either on a line, or in a rectangular auditorium with a fixed number of columns but an arbitrary number of rows, that obey any prescribed set of ‘social distancing’ restrictions. In addition to enumeration, we study the statistical distribution of the density, and give simulation algorithms for generating them.

We discuss the use of machine learning models for finding “good coordinates” for polynomial ideals. Our main goal is to put ideals into quasi-stable position, as this generic position shares most properties of the generic initial ideal, but can be deterministically reached and verified. Furthermore, it entails a Noether normalisation and provides us with a system of parameters. Traditional approaches use either random choices which typically destroy all sparsity or rather simple human heuristics which are only moderately successful. Our experiments show that machine learning models provide us here with interesting alternatives that most of the time make nearly optimal choices.

In this paper, we present an algorithm that checks if a second-order differential operator $L\in C\left(x\right)[\partial ]$ can be reduced to the general Heun's differential operator. The algorithm detects the parameters of the transformations in $C\left(x\right)[\partial ]$ that transfer the general Heun's differential operator to the operator L whose solutions are of the form(1)$\exp (\int rdx)\cdot \mathrm{HeunG}(a,q;\alpha ,\beta ,\gamma ,\delta ;f(x\left)\right),$ where $\{\alpha ,\beta ,\delta ,\gamma \}\in Q\setminus Z$, the functions $r,f\left(x\right)\in C\left(x\right)$, and $C\left(f\right(x\left)\right)$ is a subfield of $C\left(x\right)$ of index 2 or 3 or $f\left(x\right)=\frac{a{x}^n+b}{c{x}^n+d}$ for some n in $N\setminus \left\{1\right\}$.

In this paper, we present an algorithm to compute the deviation of the cranks from the average by using the theory of modular forms and Jacobi forms. Then applying the Ramanujan-type algorithm developed by Chen, Du and Zhao to each term in the expression of the deviation, we can derive the corresponding dissection formulas. As applications, we revisit the deviation of the cranks modulo 5 and 7, which were given by Garvan, and Mortenson, and also obtain the deviation of the cranks modulo 9 and 14.

In this paper we present a key exchange protocol in which Alice and Bob have secret keys given by two conics embedded in a large ambient space by means of the Veronese embedding and public keys given by hyperplanes containing the embedded curves. Both of them construct some common invariants given by the intersection of two conics.

This paper proposes ideas to speed up the process of creative telescoping, particularly when the telescoper is reducible. One can interpret telescoping as computing an annihilator $L\in D$ for an element m in a D-module M. The main idea in this paper is to look for submodules of M. If N is a non-trivial submodule of M, constructing the minimal annihilator R of the image of m in $M/N$ gives a right-factor of L in D. Then $L=L^{\prime }R$ where the left-factor $L^{\prime }$ is the telescoper of $R\left(m\right)\in N$. To expedite computing $L^{\prime }$, compute the action of D on a natural basis of N, then obtain $L^{\prime }$ with a cyclic vector computation.

The next main idea is to construct submodules from automorphisms, if we can find some. An automorphism with distinct eigenvalues can be used to decompose N as a direct sum $N_1\oplus \cdots \oplus N_k$. Then $L^{\prime }$ is the LCLM (Least Common Left Multiple) of $L_1,\dots ,L_k$ where $L_i$ is the telescoper of the projection of $R\left(m\right)$ on $N_i$. An LCLM can greatly increase the degrees of coefficients, so $L^{\prime }$ and L can be much larger expressions than the factors $L_1,\dots ,L_k$ and R. Examples show that computing each factor $L_i$ and R separately can save a lot of CPU time compared to computing L in expanded form with standard creative telescoping.

We propose a way to split a given bivariate P-recursive sequence into a summable part and a non-summable part in such a way that the non-summable part is minimal in some sense. This decomposition gives rise to a new reduction-based creative telescoping algorithm based on the concept of integral bases.

We develop a new eigenvalue method for solving structured polynomial equations over any field. The equations are defined on a projective algebraic variety which admits a rational parameterization by a Khovanskii basis, e.g., a Grassmannian in its Plücker embedding. This generalizes established algorithms for toric varieties, and introduces the effective use of Khovanskii bases in computer algebra. We investigate regularity questions and discuss several applications.

Creative telescoping is an algorithmic method initiated by Zeilberger to compute definite sums by synthesizing summands that telescope, called certificates. We describe a creative telescoping algorithm that computes telescopers for definite sums of D-finite functions as well as the associated certificates in a compact form. The algorithm relies on a discrete analogue of the generalized Hermite reduction, or equivalently, a generalization of the Abramov-Petkovšek reduction. We provide a Maple implementation with good timings on a variety of examples.