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Root Separation


The root separation (or zero separation) of a polynomial P(x) with roots r_1, r_2, ... is defined by

 Delta(P)=min_(i!=j)|r_i-r_j|.

There are lower bounds on how close two distinct roots of a polynomial can be. In particular, if P(x) is a squarefree polynomial of degree d with discriminant D, then the Mahler bound gives the minimum separation distance between any pair of roots as

 Delta(p)>sqrt((3|D|)/(d^((d+2))))||p||_2^(1-d).

See also

Polynomial Discriminant, Root, Root Isolation

This entry contributed by Bhuvanesh Bhatt

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References

Mahler, K. "An Inequality for the Discriminant of a Polynomial." Michigan Math. J. 11, 257-262, 1964.Zippel, R. Effective Polynomial Computation. Boston, MA: Kluwer, pp. 186-187, 1993.

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Root Separation

Cite this as:

Bhatt, Bhuvanesh. "Root Separation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/RootSeparation.html

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