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A Measure for Polynomials in Several Variables

Published online by Cambridge University Press:  20 November 2018

Gerald Myerson
Gerald Myerson*
Affiliation:
University of British Columbia, VancouverB.C.
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Abstract

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We define a notion of measure for polynomials in several variables, basing our construction on the geometry of the zero-set of the polynomial. For polynomials in one variable, this measure reduces to the usual one. We begin the development of the theory of this measure along lines parallel to the theory of Mahler’s measure, indicating the differences and similarities between the two.

Let P be a polynomial in one variable with complex coefficients, P(z) = a0Π(z - αj). The measure of P, denoted M(P), is defined by

1

As is well-known, an application of Jensen’s formula yields

2

Keywords

10F3532A3030C15Mahler’s measureintegrals over spheresKronecker’s theorem
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 2 , 01 June 1984 , pp. 185 - 191
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Boyd, D. W., Variations on a theme of Kronecker, Canad. Math. Bull, 21 (1978) 129–133. MR58 #5580.Google Scholar
2. Boyd, D. W., Reciprocal polynomials having small measure, Math. Comp. 35 (1980) 1361–1377. MR82a: 30005.Google Scholar
3. Boyd, D. W., Kronecker’s theorem and Lehmefs problem for polynomials in several variables, J. Number Theory 13 (1981) 116–121. MR82e: 10066.Google Scholar
4. Clarke, B., Asymptotes and intercepts of real-power polynomial surfaces from the geometry of the exponent polytope, SIAM J. Appl. Math. 35 (1978) 755–786. MR80m: 52003.Google Scholar
5. Coxeter, H. S. M., Twelve geometric essays, SIU Pr., Carbondale 1968. MR46 #9843. Reprinted from Quart J. Math. 6 (1935) 13–29.Google Scholar
6. Lawton, W. M., A generalization of a theorem of Kronecker, J. Sci. Fac. Chiangmai U. (Thailand) 4 (1977) 15–23.Google Scholar
7. Lehmer, D. H., Factorization of certain cyclotomic functions, Ann. of Math. 34 (1933) 461–479.Google Scholar
8. Mahler, K., On some inequalities for polynomials in several variables, J. London Math. Soc. 37 (1962) 341–344. MR25 #2036.Google Scholar
9. Mahler, K., Lectures on transcendental numbers, Lecture notes in Math. 546, Springer 1976. MR58 #10772.Google Scholar
10. Milnor, J., Hyperbolic geometry: the first 150 years, Bull. Amer. Math. Soc. 6 (1982) 9–24. MR82m: 57005.Google Scholar
11. Montgomery, H. L. and Schinzel, A., Some arithmetic properties of polynomials in several variables, in Transcendence Theory; Advances and Applications, Baker, A. and Masser, D. W., eds. Acad Pr 1977. MR57 #12447.Google Scholar
12. Myerson, G., A combinatorial problem in finite fields, II, Quart. J. Math. 31 (1980) 219–231. MR81i: 05014.Google Scholar
13. Myerson, G. and Smyth, C. J., On measures of polynomials in several variables: Corrigendum, Bull. Austral. Math. Soc. 26 (1982) 317–319.Google Scholar
14. Rudin, W., Function theory in the unit ball of ℂn , Springer 1980. MR82i: 32002.Google Scholar
15. Smyth, C. J., A Kronecker-type theorem for complex polynomials in several variables, Canad. Math. Bull. 24 (1981) 447–452.Google Scholar