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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Abstract | References | Similar Articles | Additional Information

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 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 has a sparse square, and explain how the substantial computation was effected using the system CoCoA.

**1.**William W. Adams and Philippe Loustaunau,*An introduction to Gröbner bases*, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence, RI, 1994. MR**1287608****2.**Don Coppersmith and James Davenport,*Polynomials whose powers are sparse*, Acta Arith.**58**(1991), no. 1, 79–87. MR**1111092****3.**A Capani, G Niesi, L Robbiano,*CoCoA: Computations in Commutative Algebra*`http://cocoa.dima.unige.it/`

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Additional Information

**John Abbott**

Affiliation:
Dipartimento di Matematica, Università di Genova, Italy

Email:
abbott@dima.unige.it

DOI:
http://dx.doi.org/10.1090/S0025-5718-00-01294-1

Received by editor(s):
February 1, 2000

Published electronically:
October 17, 2000

Article copyright:
© Copyright 2000
American Mathematical Society