# GaloisGroupFactorizationUtilities(R, UP, F)¶

GaloisGroupFactorizationUtilities provides functions that will be used by the factorizer.

beauzamyBound: UP -> Integer
beauzamyBound(p) returns a bound on the larger coefficient of any factor of p.
bombieriNorm: (UP, PositiveInteger) -> F
bombieriNorm(p, n) returns the nth Bombieri's norm of p.
bombieriNorm: UP -> F
bombieriNorm(p) returns quadratic Bombieri's norm of p.
height: UP -> F
height(p) returns the maximal absolute value of the coefficients of the polynomial p.
infinityNorm: UP -> F
infinityNorm(f) returns the maximal absolute value of the coefficients of the polynomial f.
length: UP -> F
length(p) returns the sum of the absolute values of the coefficients of the polynomial p.
norm: (UP, PositiveInteger) -> F
norm(f, p) returns the lp norm of the polynomial f.
quadraticNorm(f) returns the l2 norm of the polynomial f.
rootBound(p) returns a bound on the largest norm of the complex roots of p.
singleFactorBound(p, r) returns a bound on the infinite norm of the factor of p with smallest Bombieri's norm. r is a lower bound for the number of factors of p. p shall be of degree higher or equal to 2.
singleFactorBound(p) returns a bound on the infinite norm of the factor of p with smallest Bombieri's norm. p shall be of degree higher or equal to 2.