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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The number of solutions to linear Diophantine equations and multivariate splines
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by Wolfgang Dahmen and Charles A. Micchelli PDF
Trans. Amer. Math. Soc. 308 (1988), 509-532 Request permission

Abstract:

In this paper we study how the number of nonnegative integer solutions of $s$ integer linear equations in $n \geqslant s$ unknowns varies as a function of the inhomogeneous terms. Aside from deriving various recurrence relations for this function, we establish some of its detailed structural properties. In particular, we show that on certain subsets of lattice points it is a polynomial. The univariate case ($s = 1$) yields E. T. Bell’s description of Sylvester’s denumerants. Our approach to this problem relies upon the use of polyhedral splines. As an example of this method we obtain results of R. Stanley on the problem of counting the number of magic squares.
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 308 (1988), 509-532
  • MSC: Primary 11D04; Secondary 41A15
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0951619-X
  • MathSciNet review: 951619