Search results
Results from the WOW.Com Content Network
A product of monic polynomials is monic. A product of polynomials is monic if and only if the product of the leading coefficients of the factors equals 1. This implies that, the monic polynomials in a univariate polynomial ring over a commutative ring form a monoid under polynomial multiplication. Two monic polynomials are associated if and ...
The number of irreducible monic polynomials of degree n over F q is the number of aperiodic necklaces, given by Moreau's necklace-counting function M q (n). The closely related necklace function N q (n) counts monic polynomials of degree n which are primary (a power of an irreducible); or alternatively irreducible polynomials of all degrees d ...
In linear algebra, the minimal polynomial μ A of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with Q(A) = 0 is a (polynomial) multiple of μ A. The following three statements are equivalent: λ is a root of μ A, λ is a root of the characteristic polynomial χ A ...
Applied to the monic polynomial + = with all coefficients a k considered as free parameters, this means that every symmetric polynomial expression S(x 1,...,x n) in its roots can be expressed instead as a polynomial expression P(a 1,...,a n) in terms of its coefficients only, in other words without requiring knowledge of the roots.
For example, if x 2 − x − 1 = 0, y 3 − y − 1 = 0 and z = xy, then eliminating x and y from z − xy = 0 and the polynomials satisfied by x and y using the resultant gives z 6 − 3z 4 − 4z 3 + z 2 + z − 1 = 0, which is irreducible, and is the monic equation satisfied by the product.
He pointed out that the classical Kronecker's theorem, which characterizes monic polynomials with integer coefficients all of whose roots are inside the unit disk, can be regarded as characterizing those polynomials of one variable whose measure is exactly 1, and that this result extends to polynomials in several variables. [6]
For every n, the polynomial () is monic, has integer coefficients, and is irreducible over the integers and the rational numbers. All its roots are real ; they are the real numbers 2 cos ( 2 k π / n ) {\displaystyle 2\cos \left(2k\pi /n\right)} with k {\displaystyle k} coprime with n {\displaystyle n} and either 1 ≤ k < n {\displaystyle ...
There are a little more than 50 links to Monic polynomial. Interestingly, there are 343 occurrences of "monic" in articles, and a quick sampling indicates that roughly half of them are to "monic as in monic polynomial". (This is true for 22 of the first 40 occurrences.