Pure product polynomials and the Prouhet-Tarry-Escott problem
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- by Roy Maltby PDF
- Math. Comp. 66 (1997), 1323-1340
Abstract:
An $n$-factor pure product is a polynomial which can be expressed in the form $\prod _{i=1}^n(1-x^{\alpha _i})$ for some natural numbers $\alpha _1,\ldots ,\alpha _n$. We define the norm of a polynomial to be the sum of the absolute values of the coefficients. It is known that every $n$-factor pure product has norm at least $2n$. We describe three algorithms for determining the least norm an $n$-factor pure product can have. We report results of our computations using one of these algorithms which include the result that every $n$-factor pure product has norm strictly greater than $2n$ if $n$ is $7$, $9$, $10$, or $11$.References
- George Bachman and Lawrence Narici, Functional analysis, Academic Press, New York-London, 1966. MR 0217549
- L. Bastien, Impossibilité de $u+v\: {{3}\atop {{=}\atop {\ }}}\: x+y+z$, Sphinx-Oedipe 8 (1913), 171–172.
- E. Bombieri and J. Vaaler, On Siegel’s lemma, Invent. Math. 73 (1983), no. 1, 11–32. MR 707346, DOI 10.1007/BF01393823
- Peter Borwein, personal communication.
- Peter Borwein and Colin Ingalls, The Prouhet-Tarry-Escott problem revisited, Enseign. Math. (2) 40 (1994), no. 1-2, 3–27. MR 1279058
- H.L. Dorwart and O.E. Brown, “The Tarry-Escott Problem”, M.A.A. Monthly 44 (1937), 613–626.
- P. Erdős and G. Szekeres, On the product $\Pi ^n_{k=1}(1-z^ak)$, Acad. Serbe Sci. Publ. Inst. Math. 13 (1959), 29–34. MR 126425
- E. Laguerre, Oeuvres, vol. 1, Gauthier-Villars, Paris (1898).
- Samuel J. Maltby, “Some Optimal Results Related to the PTE Problem”, preprint.
- Samuel J. Maltby, personal communication.
- L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
- R. Maltby, Pure Product Polynomials of Small Norm, Ph.D. dissertation, Simon Fraser University (1996).
- R. Maltby “Root Systems and the Erdős-Szekeres Problem”, submitted to Acta Arithmetica.
- G. Pólya, “Sur un théorème de Laguerre”, C.R. Acad. Sci. Paris 156 (1913), 996–999.
- C.L. Siegel, “Über einige Anwendungen diophantischer Approximationen”, Abhandlungen der Preussischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse, Nr. 1 (1929), (Gesammelte Abhandlungen I, 209–266, Springer-Verlag, 1966).
- John Steinig, On some rules of Laguerre’s, and systems of equal sums of like powers, Rend. Mat. (6) 4 (1971), 629–644 (1972) (English, with Italian summary). MR 309867
- David Walley, personal communication.
Additional Information
- Roy Maltby
- Affiliation: Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6
- Email: maltby@cecm.sfu.ca
- Received by editor(s): October 16, 1995
- Received by editor(s) in revised form: June 19, 1996
- © Copyright 1997 by the author
- Journal: Math. Comp. 66 (1997), 1323-1340
- MSC (1991): Primary 11Y50, 11B75
- DOI: https://doi.org/10.1090/S0025-5718-97-00865-X
- MathSciNet review: 1422792