
	Where results make sense

Topic: Minimal polynomial

Related Topics

Trisecting the angle

Zhuyin Fuhao

Elaine Shaffer

Bradyon

Doubling the cube

West Hanover Township, Pennsylvania

Pterophoridae

Swainsboro, Georgia

Hungarian Wirehaired Vizsla

Fair game

Vienna Genesis

Jastrzebie Zdroj

Fecal occult blood

Lufthansa Airlines

Ninoy Aquino International Airport

In the News (Mon 21 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Minimal polynomial - Wikipedia, the free encyclopedia
		In mathematics, the minimal polynomial of an object α is the monic polynomial p of least degree such that p(α)=0.
		λ is a root of the characteristic polynomial of A,
		That the minimal polynomial always divides the characteristic polynomial is a consequence of the Cayley–Hamilton theorem.
	en.wikipedia.org /wiki/Minimal_polynomial (262 words)

	Polynomial - Wikipedia, the free encyclopedia
		Because of their simple structure, polynomials are easy to evaluate, and are used extensively in numerical analysis for polynomial interpolation or to numerically integrate more complex functions.
		The degree of a term in a polynomial is the sum of all of the exponents on the variables in that term, where a variable with no exponent is understood to have an exponent of 1.
		In knot theory the Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial are important knot invariants.
	en.wikipedia.org /wiki/Polynomial (2422 words)

	PlanetMath: minimal polynomial (endomorphism)
		The minimal polynomial is intimately related to the characteristic polynomial for
		Cross-references: diagonal matrix, algebraic multiplicity, between, difference, lemma, characteristic polynomial, finite dimensional, fundamental theorem of algebra, corollary, basis, factors, eigenvalues, roots, properties, minimal polynomial, polynomials, division algorithm, degree, minimal, vectors, dimension, monic polynomial, vector space, Endomorphism
		This is version 7 of minimal polynomial (endomorphism), born on 2002-11-21, modified 2006-06-02.
	planetmath.org /encyclopedia/MinimalPolynomialEndomorphism.html (196 words)

	Learn more about Polynomial in the online encyclopedia. (Site not responding. Last check: 2007-11-06)
		Note that the polynomials of degree ≤ n are precisely those functions whose (n+1)st derivative is identically zero.
		The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Weierstrass approximation theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial.
		Depending on the degree of the polynomial to be considered, simply checking if the polynomial has linear factors can eliminate several cases, and then resorting to checking divisibility of some other irreducible polynomials, however Eisenstein's criterion can be used to more efficiently determine irreducibility.
	www.onlineencyclopedia.org /p/po/polynomial.html (1534 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		What this means is that if we consider all the polynomials with integer coefficients which have x as a root, then the minimal polynomial (something like that) is the one with the smallest degree.
		Here, z-1 is the minimal polynomial of the 1st root of unity (1), z+1 is the minimal polynomial for the primitive 2nd root of unity (-1), and z^2+z+1 is the minimal polynomial for the primitive 3rd roots of unity (-1/2 +/- i*sqrt(3)/2).
		Anyway, the minimal polynomial of a primitive n-th root of unity has degree Phi(n), which we wish to be a power of 2.
	www.math.niu.edu /~rusin/papers/known-math/99/constructible (765 words)

	Element Operations
		The minimal polynomial of the element a of the field F, relative to the ground field of F. This is the unique minimal-degree monic polynomial with coefficients in the ground field, having a as a root.
		The minimal polynomial of the element a of the field F, relative to the subfield E of F. This is the unique minimal-degree monic polynomial with coefficients in E, having a as a root.
		Given an element a of a finite field F, return the characteristic polynomial of a with respect to the subfield E of F. (This polynomial is the characteristic polynomial of the companion matrix of a written as a polynomial over E, and is a power of the minimal polynomial over E.)
	www.math.wisc.edu /help/magma/text372.html (1180 words)

	Specification of Algebraic Test function (Site not responding. Last check: 2007-11-06)
		In general the output is a multiple of the minimal polynomial of the algebraic number (i.e.
		On the other hand returning a multiple of the minimal polynomial with small norm sometimes helps: for example returning x^n-1 may be more informative than the corresponding cyclotomic factor.
		The minimal polynomial has degree 6, but if we specify the degree to be 7, the algorithm returns a reducible polynomial.
	oldweb.cecm.sfu.ca /projects/IntegerRelations/algtest.html (258 words)

	PlanetMath: minimal polynomial
		is the unique, monic non-zero polynomial such that
		Cross-references: irreducible, divisible, polynomial, monic, finite field extension
		This is version 2 of minimal polynomial, born on 2002-12-27, modified 2005-05-09.
	planetmath.org /encyclopedia/MinimalPolynomial.html (46 words)

	BCH code - Wikipedia, the free encyclopedia
		To detect errors a check polynomial can be constructed so the receiving end can detect if some errors had occurred.
		Now in GF(16) we have 15 nonzero elements, and thus our polynomial will be of degree 14 with 8 check and 7 information bits - we have 8 check bits since we have (*).
		We may also use Berlekamp-Massey algorithm for determining the error locator polynomial, and hence solve the BCH decoding problem.
	en.wikipedia.org /wiki/BCH_code (662 words)

	a question on minimal polynomial (LA)
		What the minimal polynomial must divide the characteristic polynomial, it is not the only polynomial to do so.
		For example, if the characteristic polynomial is (t-1)(t-2), both t-1 and t-2 divide the characteristic polynomial but neither is the minimal polynomial ((t-1)(t-2) itself is).
		To prove a given polynomial is the minimal polynomial you must show that it satisfies the definition of minimal polynomial: that it is the monic polynomial of lowest degree satisfied by A. (which is not the same as "f(t) is m.p.
	www.physicsforums.com /showthread.php?p=1000308#post1000308 (480 words)

	Element Operations
		If the GCD is non-trivial, then this forces a splitting of the defining polynomial, all elements of the field are reduced, and the original element may now be deemed to be zero (it may not be zero because the cofactor of the GCD may be used to perform the simplification).
		Return the minimal polynomial of the element a of the field A, relative to the base field of A. This is the unique minimal-degree irreducible monic polynomial with coefficients in the base field, having a as a root.
		Thus the illusion of a true field is sustained by forcing the minimal polynomial of a to be irreducible, by first performing whatever simplifications of A are necessary for this.
	www.umich.edu /~gpcc/scs/magma/text679.htm (909 words)

	IntegerRelations
		Degree field is to set the the maximal degree of the minimal polynomial.
		Minpoly precision field is to set the accuracy of the minimal polynomial algorithm.
		Check precision field is to set separately the accuracy checking the output of the minimal polynomial algorithm.
	oldweb.cecm.sfu.ca /projects/IntegerRelations (240 words)

	Methods and Algorithms
		are represented by polynomials whose degrees are less than the degree of the minimal polynomial of
		and addition and multiplication are performed using polynomial multiplication and addition modulo the minimal polynomial.
		uses the norm to factor a squarefree polynomial whose coefficients belong to an algebraic number field.
	www.mcs.drexel.edu /~krandick/saclib/node66.html (583 words)

	[No title]
		Suppose T is a linear operator on a finite dimensional vector space V over the field F. Prove: If r is a root for the characteristic polynomial of T then r is a root of the minimal polynomial of T. Suppose A and B are nxn matrices.
		Find the minimal polynomial for T. Find the characteristic value(s) of T Is there an ordered basis, B, for R2 of characteristic vectors for T? If YES, find such a basis,B, and find M B (T).
		Find the minimal polynomial for T. Find any characteristic value(s) of T. Is there an ordered basis, B, for R 2 of characteristic vectors for T? If YES, find such a basis,B, and M B (T) for that basis.
	www.humboldt.edu /~mef2/Courses/M344SampleFinalProblems.doc (935 words)

	Cayley-Hamilton Theorem
		is the characteristic polynomial of the matrix A and the zeros of this polynomial are
		is divisible by the minimal polynomial of the matrix A without remainder.
		be the characteristic polynomial and the minimal polynomial of the matrix A, respectively.
	www.cs.ut.ee /~toomas_l/linalg/lin1/node19.html (232 words)

	[No title]
		The Conway polynomials are also used in data bases like the Modular Atlas character tables, this was the original motivation for their definition.
		The computation method with computing the minimal polynomials of all compatible elements was rediscovered in L.S.Heath and N.A.Loehr, New algorithms for generating Conway polynomials* over finite fields, Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (Baltimore, MD, 1999), 429--437, ACM, New York (1999) There are basically two methods to compute Conway polynomials*.
		Its first argument should be an element x of a finite field F. The second argument should be a subfield K of F. It returns the minimal polynomial of x over K. If there is only one argument, K defaults to the prime subfield of F. discrete_log This symbol represents a binary function.
	www.win.tue.nl /~amc/oz/om/cds/finfield1.html (741 words)

	Hurwitz polynomial: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-06)
		A Hurwitz polynomial is a polynomial[For more, click on this link] whose coefficients are positive real number[Click link for more facts about this topic]s and whose zeros are located in the left half-plane of the complex plane complex number quick summary:
		:note: the term legendre polynomials is sometimes used (wrongly) to indicate the associated legendre polynomials....
		A cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power....
	www.absoluteastronomy.com /encyclopedia/h/hu/hurwitz_polynomial.htm (430 words)

	GMRES And The Minimal Polynomial - CAMPBELL, IPSEN, KELLEY, MEYER (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		GMRES And The Minimal Polynomial - CAMPBELL, IPSEN, KELLEY, MEYER (ResearchIndex)
		The bounds are derived by constructing residual polynomials that are related to the minimal polynomial of A and can be interpreted as follows: If the eigenvalues of A consist of a single cluster plus outliers then the convergence factor is bounded by the...
		11.8%: GMRES and the Minimal Polynomial - Campbell, Ipsen, Kelley, Meyer (1996)
	citeseer.ist.psu.edu /145026.html (473 words)

	Jordan Form and Minimal Polynomials (Site not responding. Last check: 2007-11-06)
		The minimal polynomial m(x) of a matrix A is the monic (leading coefficient=1) polynomial of least degree such that m(A)= the zero matrix.
		Lemma: If A is an nxn matrix and p(x) is any polynomial, then the eigenvalues of p(A) are p(lambda), where lambda is an eigenvalue of A. The proof is not hard, but we will try to justify the lemma with an example instead.
		Note that by the Cayley-Hamilton Theorem [if c(x) is the characteristic polynomial of A, c(A)=0], the minimal polynomial divides the characteristic polynomial.
	www.ma.iup.edu /projects/CalcDEMma/JCF/jcf06.html (180 words)

	Reciprocal polynomial - Wikipedia, the free encyclopedia
		In mathematics, for a polynomial p with complex coefficients,
		If p(z) is the minimal polynomial of z
		A consequence is that the cyclotomic polynomials Φ
	en.wikipedia.org /wiki/Reciprocal_polynomial (81 words)

	MUG: minimal polynomial for algebraic number (22.3.96)
		One of the irreducible factors of F is the sought minimal polynomial.
		Hence F is irreducible and consequently the minimal polynomial of x.
		However, if F would have been the product of two irreducible factors of about degree 60, I would have had a hard time in finding out which one was the minimal polynomial.
	www.math.rwth-aachen.de /mapleAnswers/html/126.html (515 words)

	Example 1 (Site not responding. Last check: 2007-11-06)
		The minimal polynomial also has a zero at each eigenvalue (though possibly with a smaller multiplicity).
		By default (and the Cayley-Hamilton Theorem), the minimal polynomial is the characteristic polynomial (suitably written to be monic).
		Compare the size of the Jordan block for each eigenvalue with the multiplicity of that eigenvalue in the MINIMAL polynomial.
	www.ma.iup.edu /projects/CalcDEMma/JCF/jcf9.html (187 words)

	ipedia.com: Polynomial Article (Site not responding. Last check: 2007-11-06)
		In algebra, a polynomial function, or polynomial for short, is a function of the form where x is a scalar -valued variable, n is a nonnegative integer, and a 0,..., a n are fixed scalars, called the c...
		Formulas for the roots of polynomials of degree up to 4 have been known since the sixteenth century (see quadratic equation, Gerolamo Cardano, Niccolo Fontana Tartaglia).
		In 1824, Niels Henrik Abel proved the striking result that there can be no general formula (involving only the arithmetical operations and radicals) for the roots of a polynomial of degree ≥ 5 in terms of its coefficients (see Abel-Ruffini theorem).
	www.ipedia.com /polynomial.html (1572 words)

	Element Operations
		We use the minimal polynomial to determine the answer, which means that the calculation of the maximal order is not triggered if it is not known yet.
		Let P be the minimal polynomial of a over Z, with leading coefficient a_0 and roots alpha_1,..., alpha_n.
		Given an element a from an algebraic field or order L, returns the minimal polynomial of the element over R if given otherwise the subfield or suborder F where F is the field or order over which L is defined as an extension.
	www.math.lsu.edu /magma/text634.htm (2215 words)

	H.html
		Given a sufficiently accurate floating point approximation to an algebraic number, we can compute its minimal polynomial by using a lattice basis reduction algorithm.
		This usually requires as many digits as there are in the largest coefficient (in absolute value) of the minimal polynomial multiplied by the number of terms in the minimal polynomial (the degree + 1).
		We are also trying to improve the speed of computing the greatest common divisor of two polynomials, in the case where the coefficients contain algebraic numbers.
	www.cecm.sfu.ca /CAG/Projects/High_Precision_Numerics/H1.html (598 words)

	Minimal Polynomials
		All BCH codes will have a generating polynomial because they are cyclic codes.
		The choice of generating polynomial is directly related to the minimal polynomial of an element and the minimal polynomials of its powers in the field with which we are working.
		As said before, we will need to find minimal polynomials of elements in a field to create a generating polynomial in a BCH code.
	www-math.cudenver.edu /~rrosterm/crypt_proj/node5.html (203 words)

Try your search on: Qwika (all wikis)