The characteristic polynomial for A is Af and this is plainly also the minimal polynomial for A ( or for T ).

If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

( C ) if Af is the operator induced on Af by T, then the minimal polynomial for Af is Af.

We shall find a polynomial Af such that Af is the identity on Af and is zero on the other Af, and so that Af, etc..

Note also that if Af, then Af is divisible by the polynomial p, because Af contains each Af as a factor.

It is clear that each vector in the range of Af is in Af for if **ya is in the range of Af, then Af and so Af because Af is divisible by the minimal polynomial P.

Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.

Hence the minimal polynomial for Af is Af.

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first-degree polynomials, i.e., the case in which each Af is of the form Af.

The diagonalizable operator D and the nilpotent operator N are uniquely determined by ( A ) and ( B ) and each of them is a polynomial in T.

for since it is nilpotent, the minimal polynomial for this operator is of the form Af for some Af ; ;

Thus any non-constant entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire function.

In mathematics, Horner's method ( also known as Horner scheme in the UK or Horner's rule in the U. S .) is either of two things: ( i ) an algorithm for calculating polynomials, which consists in transforming the monomial form into a computationally efficient form ; or ( ii ) a method for approximating the roots of a polynomial.

Jerrum, Valiant, and Vazirani showed that every # P-complete problem either has an FPRAS, or is essentially impossible to approximate ; if there is any polynomial-time algorithm which consistently produces an approximation of a # P-complete problem which is within a polynomial ratio in the size of the input of the exact answer, then that algorithm can be used to construct an FPRAS.

A common misconception is that the " best " CRC polynomials are derived from either an irreducible polynomial or an irreducible polynomial times the factor, which adds to the code the ability to detect all errors affecting an odd number of bits.

As will be mentioned again below, it follows from the fundamental theorem of algebra that every non-constant polynomial with real coefficients can be written as a product of polynomials with real coefficients whose degree is either 1 or 2.

The Chebyshev polynomials T < sub > n </ sub > or U < sub > n </ sub > are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence.

The set of roots ( in an extension field ) of any polynomial over F must either contain no roots of a given irreducible polynomial p, or contain all such roots ; this is Abel's irreducibility theorem.

To say that an operator reduces degree by one means that if ' is a polynomial of degree ', then ' is either a polynomial of degree, or, in case, ' is 0.

Both equations are solvable by radicals if and only if either they are factorisable in equations of lower degrees with rational coefficients or the polynomial, named Cayley resolvent, has a rational root in z, where

If a polynomial has only one variable ( univariate polynomial ), then the terms are usually written either from highest degree to lowest degree (" descending powers ") or from lowest degree to highest degree (" ascending powers ").

An argument in favor of the first meaning is also that no obvious other notion is available to designate these values ( the term power product is in use, but it does not make the absence of constants clear either ), while the notion term of a polynomial unambiguously coincides with the second meaning of monomial.

To define a symmetric function one must either indicate directly a power series as in the first construction, or give a symmetric polynomial in n indeterminates for every natural number n in a way compatible with the second construction.

The proof of this generalization is similar to the one for the original statement, considering the reduction of the coefficients modulo ; the essential point is that a single-term polynomial over the integral domain cannot decompose as a product in which at least one of the factors has more than one term ( because in such a product there can be no cancellation in the coefficient either of the highest or the lowest possible degree ).

This can happen in two ways: either it is a reducible variety, meaning that its defining quadratic factors as the product of two linear polynomials ( degree 1 ), or the polynomial is irreducible but does not define a curve, but instead a lower-dimension variety ( a point or the empty set ); this latter can only occur over a field that is not algebraically closed, such as the real numbers.

Nevertheless, many algorithms for computing cliques have been developed, either running in exponential time ( such as the Bron – Kerbosch algorithm ) or specialized to graph families such as planar graphs or perfect graphs for which the problem can be solved in polynomial time.

In particular, Gromov's theorem and the Bass – Guivarch formula imply that the order of polynomial growth of a finitely generated group is always either an integer or infinity ( excluding for example, fractional powers ).

* ( Kronecker's Theorem ) If p is an irreducible monic integer polynomial with, then either p ( z )=

* Lehmer's conjecture asserts that if p is an irreducible integer polynomial, then there is a constant such that either or.

A cubic polynomial with real coefficients can either have three real roots, or one real root and two complex conjugate roots.

Wavefront aberrations are usually described as either a sampled image or a collection of two-dimensional polynomial terms.

It results in two polynomials, a quotient and a remainder that are characterized by the following property of the polynomials: given two polynomials a and b such that b ≠ 0, there exists a unique pair of polynomials, q, the quotient, and r, the remainder, such that a = b q + r and degree ( r ) < degree ( b ) ( here the polynomial zero is supposed to have a negative degree ).

A number a is a root of P if and only if the polynomial x − a ( of degree one in x ) divides P. It may happen that x − a divides P more than once: if ( x − a )< sup > 2 </ sup > divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that ( x − a )< sup > m </ sup > divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined ( or infinite ) multiplicity.

For polynomials in more than one variable the notion of root does not exist, and there are usually infinitely many combinations of values for the variables for which the polynomial function takes the value zero.

However for certain sets of such polynomials it may happen that for only finitely many combinations all polynomial functions take the value zero.

* The graph of the zero polynomial

where n is a natural number, the coefficients are elements of R, and X is a formal symbol, whose powers X < sup > i </ sup > are just placeholders for the corresponding coefficients a < sub > i </ sub >, so that the given formal expression is just a way to encode the sequence, where there is an n such that a < sub > i </ sub > = 0 for all i > n. Two polynomials sharing the same value of n are considered equal if and only if the sequences of their coefficients are equal ; furthermore any polynomial is equal to any polynomial with greater value of n obtained from it by adding terms in front whose coefficient is zero.

Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term a < sub > i </ sub > X < sup > i </ sup > is interpreted as a polynomial that has zero coefficients at all powers of X other than X < sup > i </ sup >.

If R is an integral domain and f and g are polynomials in R, it is said that f divides g or f is a divisor of g if there exists a polynomial q in R such that f q = g. One can show that every zero gives rise to a linear divisor, or more formally, if f is a polynomial in R and r is an element of R such that f ( r ) = 0, then the polynomial ( X − r ) divides f. The converse is also true.

However, choosing a reducible polynomial can result in missed errors, due to the ring's having zero divisors.

This has the convenience that the remainder of the original bitstream with the check value appended is exactly zero, so the CRC can be checked simply by performing the polynomial division on the received bitstream and comparing the remainder with zero.