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Bombieri Norm


The Bombieri p-norm of a polynomial

 Q(x)=sum_(i=0)^na_ix^i
(1)

is defined by

 [Q]_p=[sum_(i=0)^n(n; i)^(1-p)|a_i|^p]^(1/p),
(2)

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

 [R]_2[S]_2<=(n; m)^(1/2)[Q]_2
(3)

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 Norm

This 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 Norm

Cite this as:

O'Bryant, Kevin. "Bombieri Norm." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BombieriNorm.html

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