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2.
We give an upper bound for the solutions of the family of cubic Thue inequalities | x3+ axy2+ by3|? k when a is positive and larger than a certain value depending on b. For the case k= a+| b|+1 and a?360 b4 we show that these inequalities have only trivial solutions. For the case k= a+| b|+1 and | b|=1,2, we solve these inequalities for all a?1. Our method is based on Padé approximations using Rickert's integrals. We also use a generalization of Legendre's theorem on continued fractions. 相似文献
3.
We use the hypergeometric method to solve three families of Thue inequalities of degree 3, 4 and 6, respectively, each of which is parametrized by an integral parameter. We obtain bounds for the solutions, which are astonishingly small compared to similar results which use estimates of linear forms in logarithms. 相似文献
4.
We consider the relative Thue inequalities
5.
Let \(F(X,Y)=\sum \nolimits _{i=0}^sa_iX^{r_i}Y^{r-r_i}\in {\mathbb {Z}}[X,Y]\) be a form of degree \(r=r_s\ge 3\), irreducible over \({\mathbb {Q}}\) and having at most \(s+1\) non-zero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality $$\begin{aligned} |F(X,Y)|\le h \end{aligned}$$ is \(\ll s^2h^{2/r}(1+\log h^{1/r})\). They conjectured that \(s^2\) may be replaced by s. Let $$\begin{aligned} \Psi = \max _{0\le i\le s} \max \left( \sum _{w=0}^{i-1} \frac{1}{r_i-r_w},\sum _{w= i+1}^{s}\frac{1}{r_w-r_i}\right) . \end{aligned}$$ Then we show that \(s^2\) may be replaced by \(\max (s\log ^3s, se^{\Psi })\). We also show that if \(|a_0|=|a_s|\) and \(|a_i|\le |a_0|\) for \(1\le i\le s-1\), then \(s^2\) may be replaced by \(s\log ^{3/2}s\). In particular, this is true if \(a_i\in \{-1,1\}\). 相似文献
6.
The problem y= Ax+ c, x≧0, y≧0, ( x, y)=0 is considered, where the square real matrix A and the real vector c are the data and a solution is a pair of vectors x, y. Under certain conditions on the matrix A there exists a solution for every vector c, but it cannot be unique for every c. We prove that under these conditions the maximal number of solutions is 2
n
− 1. 相似文献
7.
Let be an irreducible cubic form with positive discriminant, and with non-trivial automorphisms. We show that the Thue equation
F( x, y) = 1 has at most three integer solutions except for a few known cases. For the proof, we use an explicitly expressed cubic
form which is equivalent to F. To obtain an upper bound for the size of solutions, we use the Padé approximation method developed in our former work. To
obtain a lower bound for the size of solutions, we use a result of R. Okazaki on gaps between solutions, which is obtained
by geometric consideration.
2000 Mathematics Subject Classification Primary—11D25, 11D59 相似文献
8.
The problem of finding the asymptotic number of solutions of the system of inequalities $$\begin{gathered} \left\| {\alpha _i q} \right\|< q^{ - \sigma _i } (i = 1,...,n), \sigma _i > 0, \hfill \\ \sigma = \sum\nolimits_{i = 1}^n {\sigma _i< c(\alpha _1 ,...,\alpha _n ), q = 1,...,N,} \hfill \\ \end{gathered}$$ is solved under the assumption that for real numbers α 1,..., α n, starting from some Q=max(q 1...,q n) the inequality holds for any real λ≥0. 相似文献
9.
In this paper, we solve a certain family of diophantine equations associated with a family of cyclic quartic number fields. In fact, we prove that for and , with square-free, the Thue equation has no integral solution except the trivial ones: . 相似文献
10.
We consider the problem of counting solutions to a trinomial Thue equation -- that is, an equation where is an irreducible form in with degree at least three and with three non-zero coefficients. In a 1987 paper J. Mueller and W. Schmidt gave effective bounds for this problem. Their work was based on a series of papers by Bombieri, Bombieri-Mueller and Bombieri-Schmidt, all concerned with the ``Thue-Siegel principle" and its relation to . In this paper we give specific numerical bounds for the number of solutions to by a somewhat different approach, the difference lying in the initial step -- solving a certain diophantine approximation problem. We regard this as a real variable extremal problem, which we then solve by elementary calculus. 相似文献
11.
In this paper, we solve a family of Diophantine equations associated with families of number fields of degree 3. In fact,
we use Baker’s method find all solutions to the Thue equation .
The author was supported partially by Purdue University North Central. 相似文献
12.
The aim of the present paper is to give a new kind of point of view in the theory of variational inequalities. Our approach makes possible the study of both scalar and vector variational inequalities under a great variety of assumptions. One can include here the variational inequalities defined on reflexive or nonreflexive Banach spaces, as well as the vector variational inequalities defined on topological vector spaces. 相似文献
13.
Continuing the recent work of the second author, we prove that the diophantine equation for has exactly 12 solutions except when , when it has 16 solutions. If denotes one of the zeros of , then for we also find all with . 相似文献
14.
Let K be a field of characteristic 0 and let ( K*) n denote the n-fold Cartesian product of K*, endowed with coordinatewise multiplication. Let Γ be a subgroup of ( K*) n of finite rank. We consider equations (*) a1x1 + … + anxn = 1 in x = ( x1xn)Γ, where a = ( a1, an)( K*) n. Two tuples a, b( K*) n are called Γ-equivalent if there is a uΓ such that b = u · a. Gy?ry and the author [Compositio Math. 66 (1988) 329-354] showed that for all but finitely many Γ-equivalence classes of tuples a( K*) n, the set of solutions of (*) is contained in the union of not more than 2 (n+1! proper linear subspaces of Kn. Later, this was improved by the author [J. reine angew. Math. 432 (1992) 177-217] to ( n!) 2n+2. In the present paper we will show that for all but finitely many Γ-equivalence classes of tuples of coefficients, the set of non-degenerate solutions of (*) (i.e., with non-vanishing subsums) is contained in the union of not more than 2 n proper linear subspaces of Kn. Further we give an example showing that 2 n cannot be replaced by a quantity smaller than n. 相似文献
16.
We propose an approach to the investigation of generalized solutions of linear operators that satisfy weakened a priori inequalities. This approach generalizes several well-known definitions of generalized solutions of operator equations. We prove existence and uniqueness theorems for a generalized solution. 相似文献
17.
The paper shows which embedded and immersed polyhedra with no more than eight vertices are nonflexible. It turns out that
all embedded polyhedra are nonflexible, except possibly for polyhedra of one of the combinatorial types, for which the problem
still remains open.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 143–165, 2006. 相似文献
18.
We consider the Cauchy problem for second-order nonlinear ordinary differential inequalities with a nonlinear memory term. We obtain blow-up results under some conditions on the initial data. We also give an application to a semilinear hyperbolic equation in a bounded domain. 相似文献
20.
This paper describes a modified Newton algorithm for solving a finite system of inequalities in a finite number of iterations.This research was supported by NSF Grant No. ENG-73-08214-AO1, by NSF-RANN Grant No. ENV-76-04264, and by the United Kingdom Science Research Council. 相似文献
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