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Norton and Sălăgean [Strong Gröbner bases and cyclic codes over a finite-chain ring, in Proc. Workshop on Coding and Cryptography, Paris, Electronic Notes in Discrete Mathematics, Vol. 6 (Elsevier Science, 2001), pp. 391-401] have presented an algorithm for computing Gröbner bases over finite-chain rings. Byrne and Fitzpatrick [Gröbner bases over Galois rings with an application to decoding alternant codes, J. Symbolic Comput. 31 (2001) 565-584] have simultaneously proposed a similar algorithm for computing Gröbner bases over Galois rings (a special kind of finite-chain rings). However, they have not incorporated Buchberger's criteria into their algorithms to avoid unnecessary reductions. In this paper, we propose the adapted version of these criteria for polynomials over finite-chain rings and we show how to apply them on Norton-Sălăgean algorithm. The described algorithm has been implemented in Maple and experimented with a number of examples for the Galois rings. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to improve the bound of complexity of the well-known algorithms on polynomial ideals having complexities polynomial in dn, where d is the maximal degree of input polynomials and n is the number of variables. Instead of this bound, we present the more accurate bound max{S, Dn} where S is the size of the input polynomials in dense representation, and D is the arithmetic mean value of the degrees of input polynomials. [ABSTRACT FROM AUTHOR]

This paper shows an algorithm which computes the law of the Lie algebra associated with the complex Lie group of n × n upper-triangular matrices with exponential elements in their main diagonal. For its implementation two procedures are used, respectively, to define a basis of the Lie algebra and the nonzero brackets in its law with respect to that basis. These brackets constitute the final output of the algorithm, whose unique input is the matrix order n. Besides, its complexity is proved to be polynomial and some complementary computational data relative to its implementation are also shown. [ABSTRACT FROM AUTHOR]

The complexity status of the minimum r-star cover problem for orthogonal polygons had been open for many years, until 2004 when Ch. Worman and J. M. Keil proved it to be polynomially tractable (Polygon decomposition and the orthogonal art gallery problem, IJCGA 17(2) (2007), 105-138). However, since their algorithm has Õ(n17)-time complexity, where Õ(·) hides a polylogarithmic factor, and thus it is not practical, in this paper we present a linear-time 3-approximation algorithm. Our approach is based upon the novel partition of an orthogonal polygon into so-called o-star-shaped orthogonal polygons. [ABSTRACT FROM AUTHOR]

Büchi's square problem asks if there exists a positive integer M such that all x1, ..., xM ∈ ℤ satisfying the equations $x_{r-2}^2-2x_{r-1}^2+x_{r}^2 = 2$ for all 3 ≤ r ≤ M must also satisfy $x_r^2 = (x+r)^2$ for some integer x and for all 1 ≤ r ≤ M. Hensley's problem asks if there exists a positive integer M such that, for any integers ν and a, if (ν + r)2 - a is a square for all 1 ≤ r ≤ M, then a = 0. It is not difficult to see that a positive answer to Hensley's problem implies a positive answer to Büchi's square problem. One can ask a more general version of Hensley's problem by replacing the square power by an nth power for any integer n ≥ 2 which is called Hensley's problem for nth powers. In this paper, we will study Hensley's problem for nth powers over function fields of any characteristic. [ABSTRACT FROM AUTHOR]

We introduce and discuss the recent Agrawal–Kayal–Saxena deterministic algorithm solving primality in polynomial time. [ABSTRACT FROM AUTHOR]