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  • 1
    Electronic Resource
    Electronic Resource
    Simultaneous Approximation in Positive Characteristic (2000)
    Springer
    Monatshefte für Mathematik 131 (2000), S. 15-28 
    Details
    ISSN: 1436-5081
    Keywords: 2000 Mathematics Subject Classification: 11J61 ; 11J13 ; Key words: Simultaneous approximation ; non-archimedean absolute values
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract.  Let be the approximation exponent of a power series α (so that when α is algebraic of degree d, then by Dirichlet’s and Liouville’s Theorems). If the characteristic is positive, q is a power of the characteristic, and are related by a fractional linear transformation with polynomial coefficients, then by respective work of Voloch and of de Mathan, there are constants such that has no solution if , and infinitely many solutions if . We will formulate and prove generalizations to simultaneous approximation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s006050070021
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  • 2
    Electronic Resource
    Electronic Resource
    Northcott's theorem on heights I. A general estimate (1993)
    Springer
    Monatshefte für Mathematik 115 (1993), S. 169-181 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Given a pointP=(a o:...:a n) of projective space ℙ n =ℙ n (A) whereA is the field of algebraic numbers, letd(P) be its degree andH(P) its absolute multiplicative height. Northcott's Theorem says that givend, n andX, there are only finitely many pointsP∈ℙ n withd(P)≦d andH(P)≦X. We will show that there are at mostc(d, n)X d(d+n) such points.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01311215
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  • 3
    Electronic Resource
    Electronic Resource
    The distribution of sublattices of ℤ m (1998)
    Springer
    Monatshefte für Mathematik 125 (1998), S. 37-81 
    Details
    ISSN: 1436-5081
    Keywords: 11H06 ; 11H99 ; 11H55 ; Distribution of lattices ; generalized upper half plane
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Lattices Λ, Λ′ are similar if one can be transformed into the other by an angle-preserving linear map. Similarity classes of lattices of rankn may be parametrized by a fundamental domainℱ of the action ofGL n (ℤ) on the generalized upper half-plane ℋ n . Given 1〈n≤m and , letN(D,T) be the number of sublattices of ℋ n which have rankn, similarity class inD, and determinant ≤T. Our most basic result will be thatN(D,T)∼c 1(m, n)μ(D)T m asT→∞ for suitable setsD, where μ is the invariant measure on ℋ n . The casen=2 had been dealt with by Roelcke and by Maass using the theory of modular forms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01489457
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  • 4
    Electronic Resource
    Electronic Resource
    Über Gitterpunkte auf gewissen Flächen (1964)
    Springer
    Monatshefte für Mathematik 68 (1964), S. 59-74 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01298826
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  • 5
    Electronic Resource
    Electronic Resource
    Metrische Sätze über simultane Approximation abhängiger Größen (1964)
    Springer
    Monatshefte für Mathematik 68 (1964), S. 154-166 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01307118
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  • 6
    Electronic Resource
    Electronic Resource
    Simultaneous approximation to algebraic numbers by elements of a number field (1975)
    Springer
    Monatshefte für Mathematik 79 (1975), S. 55-66 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A special case of the main result is as follows. Given a number fieldK a number ɛ〉0 and real or complex algebraic numbers ξ1,...,ξ n with 1, ξ1,...,ξ n linearly independent overK, there are only finitely many α=(α1,...,α n ) with components inK and with |ξ1,...,α1| whereH(α) is a suitably defined height.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01533775
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  • 7
    Electronic Resource
    Electronic Resource
    On cubic polynomials (1982)
    Springer
    Monatshefte für Mathematik 93 (1982), S. 329-348 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is shown that a system ofr homogeneous cubic equations with rational coefficients has a nontrivial solution in rational integers if the number of variables is at least (10r)5. For most such systems, an asymptotic formula holds for the numberz P of solutions whose components have modulus 〈P.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01295233
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  • 8
    Electronic Resource
    Electronic Resource
    On cubic polynomials I. Hua's estimate of exponential sums (1982)
    Springer
    Monatshefte für Mathematik 93 (1982), S. 63-74 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Hua andChen gave estimates of sums $$\sum\limits_{x = 1}^q {e(\mathfrak{F}(x))} $$ wheree(z)=e 2πiz and $$\mathfrak{F}$$ is a polynomial of the typef(x)/q wheref(x)=a k x k +...+a 1 x with integer coefficients having gcd (q, a k ,...,a 1)=1 But no good estimates hold for these sums whenq is small in comparison tok. We therefore consider here a related but different class of polynomials. Special emphasis is given to the cubic case. In subsequent papers of this series we shall deal with cubic exponential sums in many variables and withp-adic and rational zeros of systems of cubic forms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01579030
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  • 9
    Electronic Resource
    Electronic Resource
    Two questions in diophantine approximation (1976)
    Springer
    Monatshefte für Mathematik 82 (1976), S. 237-245 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The first question is about a possible variation on Dirichlet's approximation theorem for linear forms αx+βy+z, wherex,y are restricted topositive integers. The second question, which turns out to be related to the first, is about approximation to elements in a power series fieldk((t −1)) by solutions β of first order linear differential equationsxβ′+yβ+z=0, wherex,y,z are polynomials int.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01526328
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  • 10
    Electronic Resource
    Electronic Resource
    Contributions to diophantine approximation in fields of series (1979)
    Springer
    Monatshefte für Mathematik 87 (1979), S. 145-165 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The following topics are taken up: (i) the dimension of certain vector spaces defined in terms of denominations of differential polynomials, (ii) examples of series for which a Thue-Siegel-Roth type theorem holds, (iii) approximation of linear forms whose coefficients satisfy algebraic differential equations, and (iv) rational approximations to solutions of algebraic difference-differential equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01305913
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