Journal of Symbolic Computation最新文献

英文中文

Proof of some conjectural congruences involving products of two binomial coefficients

IF 0.6 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-02-27 DOI: 10.1016/j.jsc.2025.102436

Guo-Shuai Mao , Xiran Zhang

In this paper, we mainly prove the following conjectures of Z.-W. Sun: Let

p \equiv 3 (\mod 4)

be a prime. Then

\sum_{k = 0}^{p - 1} \frac{{(\begin{matrix} 2 k \\ k \end{matrix})}^{2}}{(2 k - 1) 8^{k}} \equiv - (\frac{2}{p}) \frac{p + 1}{2^{p - 1} + 1} (\begin{matrix} (p + 1) / 2 \\ (p + 1) / 4 \end{matrix}) (\mod p^{2}), 3 \sum_{k = 0}^{p - 1} \frac{(\begin{matrix} 2 k \\ k \end{matrix}) (\begin{matrix} 2 k \\ k + 1 \end{matrix})}{(2 k - 1) 8^{k}} \equiv p + (\frac{2}{p}) \frac{2 p}{(\begin{matrix} (p + 1) / 2 \\ (p + 1) / 4 \end{matrix})} (\mod p^{2}),

where

(\frac{\cdot}{p})

stands for the Legendre symbol. The necessary proofs are provided by the computer algebra software Sigma to find and verify the underlying hypergeometric sum identities.

在本文中，我们主要证明了 Z.-W.孙：设 p≡3(mod4) 是素数。则∑k=0p-1(2kk)2(2k-1)8k≡-(2p)p+12p-1+1((p+1)/2(p+1)/4)(modp2),3∑k=0p-1(2kk)(2kk+1)(2k-1)8k≡p+(2p)2p((p+1)/2(p+1)/4)(modp2)，其中 (⋅p) 表示 Legendre 符号。必要的证明由计算机代数软件 Sigma 提供，用于查找和验证基本的超几何和等式。

引用次数: 0

A propositional encoding for first-order clausal entailment over infinitely many constants

IF 0.6 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-02-19 DOI: 10.1016/j.jsc.2025.102434

Vaishak Belle

There is a fundamental trade-off between the expressiveness of the language and the tractability of the reasoning task in knowledge representation. On the one hand it is widely acknowledged that relations and more generally, the expressiveness of first-order logic is extremely useful for capturing concepts required for common-sense reasoning. But at the same time the entailment problem is only semi-decidable.

There have been a wide range of approaches to deal with this trade-off, from restricting the language to propositional logic to limit the expressiveness of the language in terms of the arity of the predicates (as in description logics) or the use of negation (as in Horn logic) to limit reasoning by weakening the entailment relation using non-standard semantics.

In this work, we address a gap in this literature. We show that there is an intuitive fragment of first-order disjunctive knowledge, for which reasoning is decidable and can be reduced to propositional satisfiability. Knowledge bases in this fragment correspond to universally quantified first-order clauses, but without arity restrictions and without restrictions on the appearance of negation. Queries, however, are expected to be ground formulas. We achieve this result by showing how the entailment over infinitely many infinite-sized structures can be reduced to a search over finitely many finite-size structures. The crux of the argument lies in showing that constants not mentioned in the knowledge base and/or query behave identically (in a suitable formal sense). We then go on to also show that there is also an extension to this result for function symbols.

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引用次数: 0

Reduction systems and degree bounds for integration

IF 0.6 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-02-19 DOI: 10.1016/j.jsc.2025.102432

Hao Du , Clemens G. Raab

In symbolic integration, the Risch–Norman algorithm aims to find closed forms of elementary integrals over differential fields by an ansatz for the integral, which usually is based on heuristic degree bounds. Norman presented an approach that avoids degree bounds and only relies on the completion of reduction systems. We give a formalization of his approach and we develop a refined completion process, which terminates in more instances. In some situations when the completion process does not terminate, one can detect patterns allowing to still describe infinite reduction systems that are complete. We present such infinite systems for the fields generated by Airy functions and complete elliptic integrals, respectively. Moreover, we show how complete reduction systems can be used to find rigorous degree bounds. In particular, we give a general formula for weighted degree bounds and we apply it to find tight bounds in the above examples.

引用次数: 0

Congruence properties for Schmidt type d-fold partition diamonds

IF 0.6 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-02-19 DOI: 10.1016/j.jsc.2025.102431

Olivia X.M. Yao, Xuan Yu

Recently, Dockery, Jameson, Sellers and Wilson introduced new combinatorial objects called d-fold partition diamonds, which generalize both the classical partition function and the plane partition diamonds of Andrews, Paule and Riese. They also investigated a partition function

s_{d} (n)

which counts the number of Schmidt type d-fold partition diamonds of n. They presented the generating functions of

s_{d} (n)

and proved several congruences for

s_{d} (n)

. At the end of their paper, they posed a conjecture on congruences modulo 7 for

s_{6 k + 1} (n)

and

s_{6 k + 2} (n)

. In this paper, we prove the conjectural congruences for

s_{6 k + 1} (n)

by using two methods: an elementary proof based on a result of Garvan and an algorithmic proof based on the Mathematica package RaduRK by Smoot. We also give an algorithmic proof of the conjectural congruences for

s_{6 k + 2} (n)

by utilizing Smoot's Mathematica package RaduRK. In addition, we prove new infinite families of congruences modulo 7 for

s_{6 k + 1} (n)

and prove that

\frac{s_{6 k + 1} (7 n + 3)}{7}

takes integer values with probability 1 for

n \geq 0

. Moreover, we show that there exist infinitely many integers

r_{i}

such that

s_{12 k + 1} (r_{i}) \equiv i (\mod 13)

with

0 \leq i \leq 12

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引用次数: 0

On other two representations of the C-recursive integer sequences by terms in modular arithmetic

IF 0.6 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-02-19 DOI: 10.1016/j.jsc.2025.102433

Mihai Prunescu

<div><div>If <span><math><mi>s</mi><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> is a sequence satisfying a recurrence rule of the form:<span><span><span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>+</mo><mi>d</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>s</mi><mo>(</mo><mi>n</mi><mo>+</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>d</mi></mrow></msub><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span></span></span> with coefficients <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>Z</mi></math></span>, then there exist <span><math><mi>b</mi><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>N</mi></math></span> such that for all <span><math><mi>n</mi><mo>≥</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> the following representations work:<span><span><span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mrow><mo>⌊</mo><mfrac><mrow><mo>[</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mo>(</mo><mi>d</mi><mo>−</mo><mn>2</mn><mo>)</mo><mo>+</mo><mo>⌈</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌉</mo></mrow></msup><mo>+</mo><mi>A</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>]</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>B</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>b</mi></mrow><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>n</mi></mrow></msup></mrow></mfrac><mo>⌋</mo></mrow><mo>,</mo></math></span></span></span><span><span><span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mo>|</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>|</mo></mrow></mfrac><mrow><mo>{</mo><mrow><mo>[</mo><mrow><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mrow><mo>⌈</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌉</mo></mrow></mrow></msup><mo>−</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msup><mi>sgn</mi><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mi>A</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow></mrow></mrow><mspace></mspace><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>B</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>]</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>}</mo><mo>.</mo></math></span></span></span> Here <span><math><mi>A</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>B</mi><mo>(</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> are polynomials with integer coefficients in <span><math><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Th

如果 s∈NN 是一个满足递推规则的序列，其形式为：s(n+d)+α1s(n+d-1)+...+αds(n)=0，系数 αi∈Z，那么存在 b,n0∈N，使得对于所有 n≥n0，以下表示有效：s(n)=⌊[bn(d-2)+⌈n/2⌉+A(b,n)]modB(b,n)b(d-1)n⌋,s(n)=1|αd|{[(bn(d-1)+⌈n/2⌉-bnsgn(αd)A(b,n))modB(b,n)]modbn}。这里的 A(b,n) 和 B(b,n) 是在 bn 中具有整数系数的多项式。它们可以写成：bn2f(b-n)=A(b,n)B(b,n)，其中有理函数 f(z) 是序列 (s(n)) 的生成函数。如果 s∈ZN，那么 s 可以表示为属于 NN 的序列（s(n)+cn+1）的上述任意表示法与几何级数 cn+1 之间的差。这里 c∈N 是一个足够大的常数。

引用次数: 0

Wilf-Zeilberger seeds and non-trivial hypergeometric identities

IF 0.6 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-01-28 DOI: 10.1016/j.jsc.2025.102421

Kam Cheong Au

We introduce a systematic approach for generating Wilf-Zeilberger-pairs, and prove some hypergeometric identities conjectured by J. Guillera, Z.W. Sun, Y. Zhao and others, including two Ramanujan-

1 / π^{4}

, one

1 / π^{3}

formulas as well as a remarkable series for

ζ (5)

引用次数: 0

First-order factors of linear Mahler operators

IF 0.6 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-01-28 DOI: 10.1016/j.jsc.2025.102424

Frédéric Chyzak , Thomas Dreyfus , Philippe Dumas , Marc Mezzarobba

We develop and compare two algorithms for computing first-order right-hand factors in the ring of linear Mahler operators

ℓ_{r} M^{r} + \dots + ℓ_{1} M + ℓ_{0}

where

ℓ_{0}, \dots, ℓ_{r}

are polynomials in x and

M x = x^{b} M

for some integer

b \geq 2

. In other words, we give algorithms for finding all formal infinite product solutions of linear functional equations

ℓ_{r} (x) f (x^{b^{r}}) + \dots + ℓ_{1} (x) f (x^{b}) + ℓ_{0} (x) f (x) = 0

The first of our algorithms is adapted from Petkovšek's classical algorithm for the analogous problem in the case of linear recurrences. The second one proceeds by computing a basis of generalized power series solutions of the functional equation and by using Hermite–Padé approximants to detect those linear combinations of the solutions that correspond to first-order factors.

We present implementations of both algorithms and discuss their use in combination with criteria from the literature to prove the differential transcendence of power series solutions of Mahler equations.

引用次数: 0

The conjugacy problem and canonical representatives in finitely generated nilpotent groups

IF 0.6 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-01-27 DOI: 10.1016/j.jsc.2025.102422

Bettina Eick, Óscar Fernández Ayala

We introduce a variation on the conjugacy problem for elements and subgroups in a finitely generated nilpotent group G given by a nilpotent presentation and we describe effective algorithms for its solution. While the classical conjugacy problem takes elements or subgroups a and b of G and asks to construct

g \in G

with

a^{g} = b

, our variation defines and determines a canonical representative

C a n o_{G} (a)

a^{G}

. This allows to solve the conjugacy problem via an equality test

C a n o_{G} (a) = C a n o_{G} (b)

. Additionally, our algorithms compute the associated centralizers or normalizers, respectively. We exhibit a variety of examples to demonstrate that our new methods are highly effective and often outperform the existing methods to solve the conjugacy problems for elements and subgroups in finitely generated nilpotent groups.

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引用次数: 0

On the existence of telescopers for P-recursive sequences

IF 0.6 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-01-27 DOI: 10.1016/j.jsc.2025.102423

Lixin Du

We extend the criterion on the existence of telescopers for hypergeometric terms to the case of P-recursive sequences. This criterion is based on the concept of integral bases and the generalized Abramov-Petkovšek reduction for P-recursive sequences.

引用次数: 0

Fast evaluation of generalized Todd polynomials: Applications to MacMahon's partition analysis and integer programming

IF 0.6 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-01-27 DOI: 10.1016/j.jsc.2025.102420

Guoce Xin , Yingrui Zhang , ZiHao Zhang

The Todd polynomials, denoted as

{td}_{k} (b_{1}, b_{2}, \dots, b_{m})

, are characterized by their generating function:

\sum_{k \geq 0} {td}_{k} s^{k} = \prod_{i = 1}^{m} \frac{b_{i} s}{e^{b_{i} s} - 1} .

These polynomials serve as fundamental components in the Todd class of toric varieties – a concept of significant relevance in the study of lattice polytopes and number theory. We identify that generalized Todd polynomials emerge naturally within the realm of MacMahon's partition analysis, particularly in the context of computing the Ehrhart series. We introduce an efficient method for the evaluation of generalized Todd polynomials for numerical values of

b_{i}

. This is achieved through the development of expedited operations in the quotient ring

Z_{p} [[s]]

modulo

s^{d}

, where p is a large prime. The practical implications of our work are demonstrated through two applications: firstly, we facilitate a recalculated resolution of the Ehrhart series for magic squares of order 6, a problem initially addressed by the first author, reducing computation time from 70 days to approximately 1 day; secondly, we present a polynomial-time algorithm for Integer Linear Programming in the scenario where the dimension is fixed, exhibiting a notable enhancement in computational efficiency.

{"title":"Fast evaluation of generalized Todd polynomials: Applications to MacMahon's partition analysis and integer programming","authors":"Guoce Xin , Yingrui Zhang , ZiHao Zhang","doi":"10.1016/j.jsc.2025.102420","DOIUrl":"10.1016/j.jsc.2025.102420","url":null,"abstract":"<div><div>The Todd polynomials, denoted as <span><math><msub><mrow><mtext>td</mtext></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span>, are characterized by their generating function:<span><span><span><math><munder><mo>∑</mo><mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></munder><msub><mrow><mtext>td</mtext></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><mfrac><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>s</mi></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>s</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow></mfrac><mo>.</mo></math></span></span></span> These polynomials serve as fundamental components in the Todd class of toric varieties – a concept of significant relevance in the study of lattice polytopes and number theory. We identify that generalized Todd polynomials emerge naturally within the realm of MacMahon's partition analysis, particularly in the context of computing the Ehrhart series. We introduce an efficient method for the evaluation of generalized Todd polynomials for numerical values of <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. This is achieved through the development of expedited operations in the quotient ring <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>[</mo><mo>[</mo><mi>s</mi><mo>]</mo><mo>]</mo></math></span> modulo <span><math><msup><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, where <em>p</em> is a large prime. The practical implications of our work are demonstrated through two applications: firstly, we facilitate a recalculated resolution of the Ehrhart series for magic squares of order 6, a problem initially addressed by the first author, reducing computation time from 70 days to approximately 1 day; secondly, we present a polynomial-time algorithm for Integer Linear Programming in the scenario where the dimension is fixed, exhibiting a notable enhancement in computational efficiency.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"130 ","pages":"Article 102420"},"PeriodicalIF":0.6,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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