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Symbolic Computation
Submissions received from Fri 14 Jan 22 to Tue 18 Jan 22, announced Wed, 19 Jan 22
Cross-lists
[ total of 3 entries: 1-3 ]
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New submissions for Wed, 19 Jan 22
[1] arXiv:2201.06940 [pdf, other]
Title: Simple algorithm for GCD of polynomials
Authors: Pasquale Nardone, Giorgio Sonnino
Subjects: Symbolic Computation (cs.SC)
Based on the Bezout approach we propose a simple algorithm to determine the {\tt gcd} of two polynomials which doesn't need division, like the Euclidean algorithm, or determinant calculations, like the Sylvester matrix algorithm. The algorithm needs only $n$ steps for polynomials of degree $n$. Formal manipulations give the discriminant or the resultant for any degree without needing division nor determinant calculation.
Cross-lists for Wed, 19 Jan 22
[2] arXiv:2201.05895 (cross-list from math.CO) [pdf, other]
Title: Zeon and Idem-Clifford Formulations of Hypergraph Problems
Authors: Samuel Ewing, G. Stacey Staples
Subjects: Combinatorics (math.CO); Symbolic Computation (cs.SC); Rings and Algebras (math.RA)
Zeon algebras have proven to be useful for enumerating structures in graphs, such as paths, trails, cycles, matchings, cliques, and independent sets. In contrast to an ordinary graph, in which each edge connects exactly two vertices, an edge (or, "hyperedge") can join any number of vertices in a hypergraph. In game theory, hypergraphs are called simple games. Hypergraphs have been used for problems in biology, chemistry, image processing, wireless networks, and more. In the current work, zeon ("nil-Clifford") and "idem-Clifford" graph-theoretic methods are generalized to hypergraphs. In particular, zeon and idem-Clifford methods are used to enumerate paths, trails, independent sets, cliques, and matchings in hypergraphs. An approach for finding minimum hypergraph transversals is developed, and zeon formulations of some open hypergraph problems are presented.
[3] arXiv:2201.07127 (cross-list from math.CO) [pdf, ps, other]
Title: Explicit formulas for concatenations of arithmetic progressions
Authors: Bertrand Teguia Tabuguia
Comments: 16 pages, 14 references
Subjects: Combinatorics (math.CO); Symbolic Computation (cs.SC)
The sequence $(Sm(n))_{n\geqslant 0}$: $1$, $12$, $123$, $\ldots$ formed by concatenating the first $n+1$ positive integers is often called Smarandache consecutive numbers. We consider the more general case of concatenating arithmetic progressions and establish formulas to compute them. Three types of concatenation are taken into account: the right-concatenation like $(Sm(n))_{n\geqslant0}$ or the concatenation of odd integers: $1$, $13$, $135$, $\ldots$; the left-concatenation like the reverse of Smarandache consecutive numbers $(Smr(n))_{n\geqslant 0}$: $1$, $21$, $321$, $\ldots$; and the concatenation of right-concatenation and left-concatenation like $1$, $121$, $12321$, $1234321$,$\ldots$ formed by $Sm(n)$ and $Smr(n-1)$ for $n\geqslant1$, with the initial term $Sm(0)$. The resulting formulas enable fast computations of asymptotic terms of these sequences. In particular, we use our implementation in the Computer Algebra System Maple to compute billionth terms of $(Sm(n))_{n\geqslant0}$ and $(Smr(n))_{n\geqslant0}$.
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