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Mathematics of Computation

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Sparse squares of polynomials

Author: John Abbott
Journal: Math. Comp. 71 (2002), 407-413
MSC (2000): Primary 11C04; Secondary 12Y05
Published electronically: October 17, 2000
MathSciNet review: 1863010
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We answer a question left open in an article of Coppersmith and Davenport which proved the existence of polynomials whose powers are sparse, and in particular polynomials whose squares are sparse (i.e., the square has fewer terms than the original polynomial). They exhibit some polynomials of degree $12$ having sparse squares, and ask whether there are any lower degree complete polynomials with this property. We answer their question negatively by reporting that no polynomial of degree less than $12$ has a sparse square, and explain how the substantial computation was effected using the system CoCoA.

References [Enhancements On Off] (What's this?)

  • 1. W Adams, P Loustaunau, An Introduction to Gröbner Bases, Graduate Studies in Mathematics 3, Amer. Math. Soc., Providence, 1994. MR 95g:13025
  • 2. D Coppersmith, J Davenport, ``Polynomials whose powers are sparse'' Acta Arithmetica 58 (1991), 79-87. MR 92h:12001
  • 3. A Capani, G Niesi, L Robbiano, CoCoA: Computations in Commutative Algebra

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Additional Information

John Abbott
Affiliation: Dipartimento di Matematica, Università di Genova, Italy

Received by editor(s): February 1, 2000
Published electronically: October 17, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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