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  • 1980-1984  (6)
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  • 1
    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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    Paper (German National Licenses)
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  • 2
    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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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    On cubic polynomials III. Systems ofp-adic equations (1982)
    Springer
    Monatshefte für Mathematik 93 (1982), S. 211-223 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study systems of equations $$\mathfrak{F}_1 = \ldots = \mathfrak{F}_r = 0,$$ where $$\mathfrak{F}_1 , \ldots ,\mathfrak{F}_r $$ are cubic forms withp-adic coefficients. Such a system has a nontrivialp-adic solution if the number of variables is at least 50,000 ·r 3. Further we will give estimates for the number of solutions of certain systems of cubic congruences.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01299298
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    On cubic polynomials II. Multiple exponential sums (1982)
    Springer
    Monatshefte für Mathematik 93 (1982), S. 141-168 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Letf(x 1,...,x s ) be a cubic polynomial with integer coefficients,q a prime power, ande(z)=e 2πiz . We are going to estimate sums $$\sum\limits_{x_1 = 1}^q { \cdot \cdot \cdot } \sum\limits_{x_S = 1}^q {e(q^{ - 1} f(x_1 ,...,x_S ))} $$ , as well as generalizations of such sums.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01301401
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Simultaneousp-adic zeros of quadratic forms (1980)
    Springer
    Monatshefte für Mathematik 90 (1980), S. 45-65 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The main result is thatr quadratic forms in more than 4r 2+4r variables possess a common nontrivialp-adic zero.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01641711
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    Paper (German National Licenses)
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  • 6
    Book
    Book
    Analytische Methoden für Diophantische Gleichungen: einf. Vorlesungen; 5 (1982)
    Basel u.a. :Birkhäuser,
    Details
    Title: Analytische Methoden für Diophantische Gleichungen: einf. Vorlesungen; 5
    Author: Schmidt, Wolfgang M.
    Publisher: Basel u.a. :Birkhäuser,
    Year of publication: 1982
    Pages: 122 S.
    Series Statement: DMV Seminar 5
    Type of Medium: Book
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