A polynomial P is annihilating or called an annihilating polynomial in linear algebra and operator theory if the polynomial considered as a function of...

minimal polynomial μA 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...

p_{A}(A)=\mathbf {0} ;} that is, the characteristic polynomial p A {\displaystyle p_{A}} is an annihilating polynomial for A . {\displaystyle A.} One use for the...

An polynomial annihilates A {\displaystyle A} if p ( A ) = 0 {\displaystyle p(A)=0} ; p {\displaystyle p} is also known as an annihilating polynomial. Thus...

obtained from the characteristic polynomial of ⁠ A {\displaystyle A} ⁠ or, more generally, from any annihilating polynomial of ⁠ A {\displaystyle A} ⁠. This...

dividing one such polynomial by another, when each has a finite order (highest exponent), results in an infinite-order polynomial. An annihilator operator, denoted...

{\displaystyle f'(x)} . A {\displaystyle A} is called an annihilating operator of f (the annihilating operators of f {\displaystyle f} form an ideal in the...

1007/s00209-003-0490-6. Lewis, D. W.; McGarraghy, S. (2000). "Annihilating polynomials, étale algebras, trace forms and the Galois number". Archiv der...

consider the family of polynomials which annihilates an operator T {\displaystyle T} . This family is an ideal in the ring of polynomials. Furthermore, it is...

{\displaystyle f_{k}(X)} is the monic polynomial of smallest degree annihilating the module, or zero if no such non-zero polynomial exists. In the first case dim...

differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the...

Alternatively, the set of polynomials that annihilate a given A form an ideal I in C[x], the principal ideal domain of polynomials with complex coefficients. The...

extending the field F (whence the "rational"), notably without factoring polynomials, this shows that whether two matrices are similar does not change upon...

is an element of the field of fractions of A that is a root of a monic polynomial with coefficients in A, then x is itself an element of A. Many well-studied...

characteristic polynomial of A. So the algebraic multiplicity is the multiplicity of the eigenvalue as a zero of the characteristic polynomial. Since any...

are met: c i {\displaystyle c_{i}} are constants. g(x) is a constant, a polynomial function, exponential function e α x {\displaystyle e^{\alpha x}} , sine...

Third, Q {\displaystyle Q} should take a polynomial in x {\displaystyle x} and p {\displaystyle p} to a "polynomial" in X {\displaystyle X} and P {\displaystyle...

commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers Z...

commutatively (which is annihilated by the derived algebra). Thus of central interest are the free commutative algebras, namely the polynomial algebras. In this...

For example, take an irreducible polynomial f ( x 1 , … , x n ) {\displaystyle f(x_{1},\ldots ,x_{n})} in a polynomial ring F [ x 1 , … , x n ] {\displaystyle...

commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ordinary integers Failed...

given binary output sequence. The algorithm will also find the minimal polynomial of a linearly recurrent sequence in an arbitrary field. The field requirement...

theory itself and for its applications, such as homological properties and polynomial identities. Commutative rings are much better understood than noncommutative...

may be computed in terms of division polynomials. Analytic torsion Arithmetic dynamics Flat module Annihilator (ring theory) Localization of a module...

given for polynomial identity rings. The notation Z(R) will be used to denote the center of a ring R. Theorem: If R is a simple polynomial identity ring...

expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial. Early major results were the Kashiwara constructibility theorem and Kashiwara...

of a torsion-free module that is not flat is the ideal (x, y) of the polynomial ring k[x, y] over a field k, interpreted as a module over k[x, y]. Any...

{\displaystyle A^{n}} is diagonalizable, then A {\displaystyle A} is annihilated by some polynomial ( x n − λ 1 ) ⋯ ( x n − λ k ) {\displaystyle \left(x^{n}-\lambda...

variety: the dimension of the affine variety defined by an ideal I in a polynomial ring R is the Krull dimension of R/I. A field k has Krull dimension 0;...

{Supp} M\cong \operatorname {Spec} (R/I)} , the vanishing locus of the polynomial f. Looking at the short exact sequence 0 → I → R → R / I → 0 {\displaystyle...