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Topic: Monic polynomial

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	Polynomial -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		Because of their simple structure, polynomials are very easy to evaluate, and are used extensively in (additional info and facts about numerical analysis) numerical analysis for (additional info and facts about polynomial interpolation) polynomial interpolation or to (additional info and facts about numerically integrate) numerically integrate more complex functions.
		Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics.
		A polynomial with one, two or three terms is called (additional info and facts about monomial) monomial, (A quantity expressed as a sum or difference of two terms) binomial or trinomial respectively.
	www.absoluteastronomy.com /encyclopedia/p/po/polynomial.htm (2191 words)

	Encyclopedia: Monic polynomial (Site not responding. Last check: 2007-11-05)
		In linear algebra, the characteristic polynomial of a square matrix encodes several important properties of the matrix.
		As there is no general closed formula to calculate the roots of a polynomial of degree 5 and higher, root-finding algorithms are used in numerical analysis to approximate the roots.
		Polynomials with coefficients in R can be added by simply adding corresponding coefficients and multiplied using the distributive law and the rules
	www.nationmaster.com /encyclopedia/Monic-polynomial (1696 words)

	PlanetMath: discriminant
		The discriminant of a given polynomial is a number, calculated from the coefficients of that polynomial, that vanishes if and only if that polynomial has one or more multiple roots.
		There are other ways to do this of course; one can look at the formal derivative of the polynomial (it will be coprime to the original polynomial if and only if that original had no multiple roots).
		The above relation is a defining one, because the right-hand side of (1) is, evidently, a symmetric polynomial, and because the algebra of symmetric polynomials is freely generated by the basic symmetric polynomials, i.e.
	planetmath.org /encyclopedia/Discriminant.html (505 words)

	Cubic equation (Site not responding. Last check: 2007-11-05)
		A cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power.
		The polynomial is the third Chebyshev polynomial normalized to the interval [-2, 2], which we use in place of [-1, 1] so as to obtain a monic polynomial.
		This procedure is precisely analogous to the definition of the cube root in terms of logarithms and exponentials, with 2 arccosh(x) in the place of ln(x), and cosh(x/2) in the place of exp(x).
	hallencyclopedia.com /Cubic_equation (1458 words)

	PlanetMath: minimal polynomial (endomorphism)
		The minimal polynomial is intimately related to the characteristic polynomial for
		Cross-references: 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 6 of minimal polynomial (endomorphism), born on 2002-11-21, modified 2004-04-30.
	planetmath.org /encyclopedia/MinimalPolynomialEndomorphism.html (164 words)

	ipedia.com: Polynomial Article (Site not responding. Last check: 2007-11-05)
		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...
		Note that the polynomials of degree ≤ n are precisely those functions whose (n+1)st derivative is identically zero.
		In order to determine function values of polynomials for given values of the variable x, one does not apply the polynomial as a formula directly, but uses the much more efficient Horner scheme instead.
	www.ipedia.com /polynomial.html (1572 words)

	Polynomials
		Since, by definition; a0 is not zero, the polynomial equation p(x) = 0 may be divided by a0 to yield the corresponding monic polynomial equation.
		Since, in a monic polynomial equation, the arithmetic mean of the roots is minus a1; the arithmetic mean of the roots in our standard polynomial equation is zero.
		Since, in a monic polynomial equation, the geometric mean of the roots is (- an)^(1 / n); the geometric mean of the roots in our standard polynomial equation is one.
	www.rism.com /Trig/polynomi.htm (870 words)

	Monic - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-05)
		monic morphism – a special kind of morphism in category theory,
		monic polynomial – a polynomial whose leading coefficient is one.
		This is a disambiguation page — a list of articles associated with the same title.
	en.wikipedia.org /wiki/Monic (85 words)

	Appendix Four, Fermat Manual for Mac OSX/Linux/Unix/Windows
		The highest precedence polynomial variable in x is replaced with the matrix [y] and the resulting expression simplified.
		When n and m are multivariable polynomials, this procedure attempts to answer quickly by substituting each polynomial variable except the highest with a constant.
		x must be a polynomial or integer, n must be a positive integer, and m must be a monic polynomial or positive integer.
	www.bway.net /~lewis/fermat/ap4w.htm (6071 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		To complete the proof, we show that if a monic polynomial with nonzero constant term has no positive zeros, then the constant term is positive - that is, the polynomial must have an even number of sign variations.
		We assume a nonconstant monic polynomial, and note that the constant term is the polynomial's value at x = 0.
		Because the polynomial is monic, it will take positive values when x is very large.
	www.math.niu.edu /~rusin/known-math/99/descartes (666 words)

	number_field (Site not responding. Last check: 2007-11-05)
		If the given polynomial is not irreducible or not monic, this results in undefined behaviour.
		You may input the generating polynomial of the number_field; however - due to the limitations of the class istream - only the verbose format for polynomials is supported and the polynomial has to start with the variable, e.g.
		This also represents the field generated by the given polynomial, however in contrast to the previous format {1, b, b^2,..., b^n-1}*T will be used as the basis of the number_field.
	www.math.psu.edu /local_doc/LiDIA/node93.html (684 words)

	An Inequality For The Norm Of A Polynomial Factor - Pritsker (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		Let p(z) be a monic polynomial of degree n, with complex coef- cients, and let q(z) be its monic factor.
		Introduction Let p(z) be a monic polynomial of degree n, with complex...
		Pritsker, An inequality for the norm of a polynomial factor, to appear in Proc.
	citeseer.ist.psu.edu /pritsker97inequality.html (495 words)

	Symmetric Pseudoprimes - Introduction (Site not responding. Last check: 2007-11-05)
		In this form the definition is applicable to any monic polynomial with integer coefficients.
		In general, we expect a polynomial with d distinct roots to represent d non-redundant primality conditions, one for each root.
		Specifically, for any given monic polynomial f with integer coefficients, we define a symmetric pseudoprime relative to f as a composite integer N such that each elementary symmetric function of the Nth powers of the roots of f is congruent (mod N) to the same function of the first powers.
	www.mathpages.com /home/kmath003/intro.htm (465 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		The polynomial is created by randomly choosing as many exponents as specified through the option Terms and then choosing random coefficients.
		We generate a univariate random polynomial in the indeterminate z, and use the default values for the other options.
		Therefore the polynomial has integer coefficients, is of degree 5, and has 6 terms.
	www.sciface.com /STATIC/DOC30/eng/polylib_randpoly.html (403 words)

	single_factor <gf_polynomial> (Site not responding. Last check: 2007-11-05)
		bigint a.extract_unit() returns the leading coefficient of a.base() and converts it to a monic polynomial (by dividing it by its leading coefficient).
		f must be monic and square-free with a non-zero constant term; otherwise the behaviour of this function is undefined.
		f must be monic and square-free, otherwise the behaviour of this function is undefined.
	www.math.psu.edu /local_doc/LiDIA/node72.html (292 words)

	A simple factorization algorithm for univariate polynomials
		The final factoring algorithm needs the input polynomial to be monic with integer coefficients (a polynomial is monic if its leading coefficient is one).
		After the steps described here the polynomial is now monic with integer coefficients, and the factorization of this polynomial can be used to determine the factors of the original polynomial p(x).
		The same extension could be made for multivariate polynomials, although setting up the initial irreducible polynomials that divide p _test(x) modulo 2 might become expensive if done on a polynomial with many variables (2^(2^m-1) trials for m variables).
	yacas.sourceforge.net /Algochapter3.html (2777 words)

	Finite fields in MAPLE (Site not responding. Last check: 2007-11-05)
		gf_eval_poly2(f,x,y,a,b,FF) evaluates a polynomial in 2 variables f at x=a,y=b in GF field G; f must have integer coefficients (a,b in GF notation, i.e., as a polynomial in 10^4 with coefficients in 0..p-1).
		This can be explored graphically by plotting the probability that a randomly choosen irreducible polynomial of a fixed degree n has maximum period, n=3,4,5,....,100.
		Though a root finding routine for polynomials in 2 variables wasn't written, it is not hard to write down some commands which do the job.
	web.usna.navy.mil /~wdj/maplestuff/finite_fields.html (1276 words)

	Closed in the Polynomial Ring
		If f(x) and g(x) are monic polynomials in s[x], and f*g lies in c[x], then f and g both lie in c[x].
		Because w is monic, this ring, which I will call v, consists of polynomials in t, with coefficients in s, up to (but not including) the degreee of w.
		A monic polynomial p makes this happen, and together, the coefficients of p exhibit finitely many indeterminants from w.
	www.mathreference.com /id-ext,poly.html (705 words)

	Math 197 (Site not responding. Last check: 2007-11-05)
		Let F be a field and let N in F[X] be a polynomial with coefficients in F of degree d.
		Show that the additive inverse of a polynomial P of degree at most d-1 is (N-P)%N. The set of polynomials of degree at most d-1, with these operations, is denoted F[X]/(N).
		Note that F is contained in F[X]/(N) as the polynomials of degree 0.
	www.math.ucla.edu /~blasius/197.1.02s/197finalhw.htm (291 words)

	[No title]
		If p(z) is not a square, q(z) is not a polynomial, and we have q(z) = r(z) + a_k z^(-k) + O(z^(-k-1)) where r(z) is a polynomial with rational coefficients and a_k \ne 0.
		Note that there is a distinction between integer polynomials and polynomials all of whose values at integers are integers (e.g.
		There is a classical theorem which states that if a polynomial with integral coefficients is an $m$th power for every integral value of its argument, then it is the $m$th power of a polynomial with integral coefficients.
	www.math.niu.edu /~rusin/known-math/98/polysq (1188 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		We say that such a polynomial $f$ is {\em primitive} if its content is 1.
		Now observe that the product of two primitive polynomials is primitive, for if not choose a prime $p$ dividing the content of their product, and reduce coefficients modulo $p,$ recall that $\mathbb Z_p[X]$ is an integral domain and obtain the required contradiction.
		Let $m \in \mathbb Q[X]$ be the minimum (rational) polynomial of $\alpha,$ so $m$ divides $h$ in $\mathbb Q[X]$ i.e.
	www.bath.ac.uk /~masgcs/courses/nt99/nt5a.txt (523 words)

	Minimal polynomial
		The minimal polynomial of an n-by-n matrix A over a field F is the monic polynomial p(x) over F of least degree such that p(A)=0.
		λ is a root of the characteristic polynomial of A,
		In field theory, given a field extension E/F and an element α of E which is algebraic over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p(α) = 0.
	www.brainyencyclopedia.com /encyclopedia/m/mi/minimal_polynomial.html (167 words)

	Methods and Algorithms
		To factor an arbitrary univariate polynomial modulo a prime, one should first obtain a similar monic polynomial by using the algorithm
		The factorization of a squarefree primitive polynomial is performed by the algorithm
		If the polynomial A to be factored has rational base coefficients then it must first be converted to an integral polynomial by multiplying A by the least common multiple of the denominators of the base coefficients and then converting the polynomial thus obtained to integral representation.
	www.mcs.drexel.edu /~krandick/saclib/node55.html (236 words)

	Local Fields
		The polynomial is a link to the same polynomial in a form which can be readily selected and pasted into other programs.
		Other subfields are given by their sample polynomials, with the exception of unramified subfields.
		Twin Algebra = a defining polynomial for the twin algebra of a sextic field.
	math.asu.edu /~jj/localfields (708 words)

	The Algebra of Polynomial Residue Classes (Site not responding. Last check: 2007-11-05)
		Theorem 11 In the algebra of polynomials modulo a polynomial
		be an ideal in the algebra of polynomials modulo
		every polynomial as the one mentioned in the previous point is in a residue class which is in the ideal.
	www.science.unitn.it /~flego/links/tesi1/node14.html (353 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		**************************************************************/ poly_t mult(const poly_t factor1, const poly_t factor2); /*************************************************************** * plus: return the sum of two polynomials * * parameters: addend1 and addend2 represent polynomials whose sum * is sought.
		**************************************************************/ poly_t plus(const poly_t addend1, const poly_t addend2); /*************************************************************** * eval: evaluate a polynomial at a given point * * parameters: poly represents a polynomial, whose value at integer * x is sought.
		To be monic means that * poly2.coefficient[poly2.degree] must be either 1 or * -1 (and your solution may assume this is true).
	www.cs.toronto.edu /~heap/Courses/242F02/A1/Solution/Polynomial.h (350 words)

	Introduction
		This polynomial, denoted g(x), is known as the
		From other examples you can see that codes can be generated by different polynomials, but only one of these is the generator polynomial, so by using the first theorem and the definition of a monic polynomial we obtain a better definition of a generator polynomial and how to find it.
		G is based on the polynomial generator g(x).
	www.math.uri.edu /~thoma/teaching/mth391_fall2004/cyclic.htm (657 words)

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