共查询到20条相似文献,搜索用时 31 毫秒
1.
Given a polynomial with integer coefficients, we calculate the density of the set of primes modulo which the polynomial has a root. We also give a simple criterion to decide whether or not the polynomial has a root modulo every non-zero integer.
2.
In this paper we prove the best possible upper bounds for the number of elements in a set of polynomials with integer coefficients all having the same degree, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients. Moreover, we prove that there does not exist a set of more than 12 polynomials with integer coefficients and with the property from above. This significantly improves a recent result of the first two authors with Tichy [A. Dujella, C. Fuchs, R.F. Tichy, Diophantine m-tuples for linear polynomials, Period. Math. Hungar. 45 (2002) 21-33]. 相似文献
3.
The paper describes a method for determining integer solutions of a homogeneous polynomial system with integer coefficients which has finitely many solutions in the projective space over the field of complex numbers under the assumption that these solutions have a certain property. 相似文献
4.
R. Nair 《Proceedings Mathematical Sciences》1997,107(4):363-376
We show that the values of a polynomial with a-adic coefficients at integer and rational prime arguments are asymptotically
distributed on the a-adic integers and that the integer parts of certain sequences known to be uniformly distributed modulo
one, are uniformly distributed on the a-adic integers. 相似文献
5.
Description of 2-integer continuous knapsack polyhedra 总被引:1,自引:0,他引:1
In this paper we discuss the polyhedral structure of several mixed integer sets involving two integer variables. We show that the number of the corresponding facet-defining inequalities is polynomial on the size of the input data and their coefficients can also be computed in polynomial time using a known algorithm [D. Hirschberg, C. Wong, A polynomial-time algorithm for the knapsack problem with two variables, Journal of the Association for Computing Machinery 23 (1) (1976) 147–154] for the two integer knapsack problem. These mixed integer sets may arise as substructures of more complex mixed integer sets that model the feasible solutions of real application problems. 相似文献
6.
7.
An algorithm is described that determines whether a given polynomial with integer coefficients has a cyclotomic factor. The algorithm is intended to be used for sparse polynomials given as a sequence of coefficient-exponent pairs. A running analysis shows that, for a fixed number of nonzero terms, the algorithm runs in polynomial time.
8.
It is proved that a polynomial in several Mahler measures with positive rational coefficients is equal to an integer if and only if all these Mahler measures are integers. An estimate for the distance between a metric Mahler measure and an integer is obtained. Finally, it is proved that the ratio of two distinct Mahler measures of algebraic units is irrational. 相似文献
9.
We prove that there exists a polynomial F(x, t) with rational coefficients, whose degree with respect to x is equal to 4,
such that for every integer a, the Galois group of the decomposition field of the polynomial F(x, a) is not the dihedral group,
but any other transitive subgroup of the group S4 can be represented as the Galois group of the decomposition field of the polynomial F(x, a) for a certain integer a. Bibliography:
1 title.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 321, 2005, pp. 275–280. 相似文献
10.
Alexander Lifshitz 《Linear and Multilinear Algebra》2019,67(7):1420-1459
This paper presents a fraction-free (FF) version of the Bistritz test to determine the zero location (ZL) of a polynomial with respect to the unit circle. The test has the property that when it is invoked on a polynomial with Gaussian or real integer coefficients, it is an efficient integer algorithm completed without fractions over the respective integral domain. The test is not restricted to integers but remains integer preserving (IP) in all possible encounters of abnormalities and singularities. We define a symmetric subresultant polynomial sequence (SSPS) for the Sylvester matrix of two symmetric polynomials. We then show that the sequence of polynomials produced by the FF test coincides with the SSPS of its first two polynomials when the test is normal and the SSPS is strongly nonsingular, or else its polynomials match the non-singular subresultant polynomial and pass over intermediate gaps of singular subresultants in an IP and efficient manner. This relationship (interesting in its own right) is used to show that the test is IP and normally attains integers of minimal size. 相似文献
11.
A trigonometric polynomial generalization to the positivity of an alternating sum of binomial coefficients is given. The
proof uses lattice paths, and identifies the trigonometric sum as a polynomial with positive integer coefficients. Some special
cases of the q -analogue conjectured by Bressoud are established, and new conjectures are given.
January 22, 1997. Date revised: July 9, 1997. 相似文献
12.
Gyan Prakash 《Journal of Number Theory》2009,129(2):428-433
We obtain a close to optimal version of the large sieve inequality with amplitudes given by the values of a polynomial with integer coefficients of degree ?2. 相似文献
13.
In this paper, we deal with some corresponding relations between knots and polynomials by using the basic properties of knot polynomials (such as, some special values of knot polynomials, the Arf invariant and derivative of knot polynomials). We give necessary and sufficient conditions that a Laurent polynomial with integer coefficients, whose breadth is less than five, is the Jones polynomial of a certain knot. 相似文献
14.
《Journal of Algebra》1999,211(1):206-224
We show that split Jordan pairs over rings without 2-torsion can be distinguished by polynomial identities with integer coefficients. In particular, this holds for simple finite-dimensional Jordan pairs over algebraically closed fields of characteristic not 2. We also generalize results of Drensky and Racine and of Rached and Racine on polynomial identities of, respectively, Jordan algebras and Jordan triple systems. 相似文献
15.
Roberto Avanzi Waldyr Dias BenitsJr Steven D. Galbraith James McKee 《Designs, Codes and Cryptography》2011,61(1):71-89
Frobenius expansions are representations of integers to an algebraic base which are sometimes useful for efficient (hyper)elliptic
curve cryptography. The normal form of a Frobenius expansion is the polynomial with integer coefficients obtained by reducing
a Frobenius expansion modulo the characteristic polynomial of Frobenius. We consider the distribution of the coefficients
of reductions of Frobenius expansions and non-adjacent forms of Frobenius expansions (NAFs) to normal form. We give asymptotic
bounds on the coefficients which improve on naive bounds, for both genus one and genus two. We also discuss the non-uniformity
of the distribution of the coefficients (assuming a uniform distribution for Frobenius expansions). 相似文献
16.
Horst Brunotte 《Periodica Mathematica Hungarica》2013,67(1):15-32
For a given real polynomial f without positive roots we study polynomials g of lowest degree such that the product gf has positive (nonnegative, respectively) coefficients. We show that for quadratic f with negative linear coefficient every such g must have positive coefficients and exhibit an easy procedure for the determination of g. If f has only integer coefficients we show that g with integer coefficients can be found. Furthermore, for some classes of polynomials f we give upper (lower, respectively) bounds for the degrees of g. 相似文献
17.
In this paper, we prove that if {a,b,c,d} is a set of four non-zero polynomials with integer coefficients, not all constant, such that the product of any two of its distinct elements plus 1 is a square of a polynomial with integer coefficients, then
(a+b−c−d)2=4(ab+1)(cd+1). 相似文献
18.
Hilbert's Tenth Problem(HTP) asks for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring Z of integers.This was finally solved negatively by Matiyasevich in 1970.In this paper we obtain some further results on HTP over Z.We prove that there is no algorithm to determine for any P(z_1,...,z_9) ∈ Z[z_1,...,z_9] whether the equation P(z_1,...,z_9)=0 has integral solutions with z_9≥0.Consequently,there is no algorithm to test whether an arbitrary polynomial Diophantine equation P(z_1,...,z_(11))=0(with integer coefficients) in 11 unknowns has integral solutions,which provides the best record on the original HTP over Z.We also prove that there is no algorithm to test for any P(z_1,...,z_(17))∈Z[z_1,...,z_(17)] whether P(z_1,...,z_(17))=0 has integral solutions,and that there is a polynomial Q(z_1,...,z_(20))∈Z[z_1,...,z_(20)] such that {Q(z_1~2,...,z_(20)~2):z_1,...,z_(20)∈Z}∩ {0,1,2,...} coincides with the set of all primes. 相似文献
19.
Kerov considered the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as a polynomial in free cumulants. Biane has proved that this polynomial has integer coefficients, and made various conjectures. Recently, Sniady has proved Biane's conjectured explicit form for the first family of nontrivial terms in this polynomial. In this paper, we give an explicit expression for all terms in Kerov's character polynomials. Our method is through Lagrange inversion.
20.
Nicolae Ciprian Bonciocat 《代数通讯》2013,41(8):3102-3122
The famous irreducibility criteria of Schönemann–Eisenstein and Dumas rely on information on the divisibility of the coefficients of a polynomial by a single prime number. In this paper, we will use some results and ideas of Dumas to provide several irreducibility criteria of Schönemann–Eisenstein–Dumas-type for polynomials with integer coefficients, criteria that are given by some divisibility conditions for their coefficients with respect to arbitrarily many prime numbers. A special attention will be paid to those irreducibility criteria that require information on the divisibility of the coefficients by two distinct prime numbers. 相似文献