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2 results

  • FROBENIUS NUMBERS ASSOCIATED WITH DIOPHANTINE TRIPLES OF $x^2+y^2=z^3$

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  • TAKAO KOMATSU, NEHA GUPTA, MANOJ UPRETI
  • Journal:
    Bulletin of the Australian Mathematical Society , First View
    Published online by Cambridge University Press:
    14 November 2024, pp. 1-10
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  • We give an explicit formula for the Frobenius number of triples associated with the Diophantine equation $x^2+y^2=z^3$, that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of $x^2+y^2=z^3$. For the equation $x^2+y^2=z^2$, the Frobenius number has already been given. Our approach can be extended to the general equation $x^2+y^2=z^r$ for $r>3$.

  • THE SHORT RESOLUTION OF A SEMIGROUP ALGEBRA

  • Part of
  • I. OJEDA, A. VIGNERON-TENORIO
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 96 / Issue 3 / December 2017
    Published online by Cambridge University Press:
    29 August 2017, pp. 400-411
    Print publication:
    December 2017
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  • This work generalises the short resolution given by Pisón Casares [‘The short resolution of a lattice ideal’, Proc. Amer. Math. Soc.131(4) (2003), 1081–1091] to any affine semigroup. We give a characterisation of Apéry sets which provides a simple way to compute Apéry sets of affine semigroups and Frobenius numbers of numerical semigroups. We also exhibit a new characterisation of the Cohen–Macaulay property for simplicial affine semigroups.