Bombieri Norm
The Bombieri p-norm of a polynomial
| [画像: Q(x)=sum_(i=0)^na_ix^i ] |
(1)
|
is defined by
![画像: [Q]_p=[sum_(i=0)^n(n; i)^(1-p)|a_i|^p]^(1/p),](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/BombieriNorm/NumberedEquation2.svg){kind=link}
where (n; i) is a binomial coefficient. The most remarkable feature of Bombieri's norm is that given polynomials R and S such that RS=Q, then Bombieri's inequality
![画像: [R]_2[S]_2<=(n; m)^(1/2)[Q]_2](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/BombieriNorm/NumberedEquation3.svg){kind=link}
holds, where n is the degree of Q, and m is the degree of either R or S. This theorem captures the heuristic that if R and S have big coefficients, then so does RS, i.e., there can't be too much cancellation.
See also
Norm, Bombieri's Inequality, Polynomial NormThis entry contributed by Kevin O'Bryant
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References
Beauzamy, B.; Bombieri, E.; Enflo, P.; and Montgomery, H. L. "Products of Polynomials in Many Variables." J. Number Th. 36, 219-245, 1990.Borwein, P. and Erdélyi, T. "Bombieri's Norm." §5.3.E.7 in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, p. 274, 1995.Reznick, B. "An Inequality for Products of Polynomials." Proc. Amer. Math. Soc. 117, 1063-1073, 1993.Referenced on Wolfram|Alpha
Bombieri NormCite this as:
O'Bryant, Kevin. "Bombieri Norm." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BombieriNorm.html