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Topic: Discriminant of a polynomial

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	Discriminant - Wikipedia, the free encyclopedia
		The discriminant of a polynomial is a number that can be easily computed from the coefficients of the polynomial and which is zero if and only if the polynomial has a multiple root.
		The discriminant of this polynomial is defined as the determinant of the (2n − 1)×(2n − 1) matrix
		The discriminant of p(x) is thus equal to the resultant of p(x) and p'(x), where p'(x) is the derivative of p(x).
	en.wikipedia.org /wiki/Discriminant_of_a_polynomial (592 words)

	Aids Discrimination -- Recommendations and Resources (Site not responding. Last check: 2007-09-30)
		The discriminant of this polynomial is defined as the determinant of the (2''n'' − 1)×(2''n'' − 1) matrix 1 ''a''
		Critics of reverse discrimination believe proponents take a situational stand where a principled stand is more appropriate, arguing that the practice replaces one form of discrimination with another.
		As a form of discrimination, reverse discrimination is considered illegal in some countries, such as the United Kingdom.
	www.becomingapediatrician.com /health/2/aids-discrimination.html (1635 words)

	[No title] (Site not responding. Last check: 2007-09-30)
		If P is a polynomial with roots ri, the product >S=Prod(ri-rj) is not well defined, since it depends on the order of the >roots -- move them around and you may replace S by its negative >(example - 2 roots => S = (r1-r2) = -(r2-r1).
		That is, the discriminant is the result of computing a certain determinant made from the polynomial coefficients.
		Discriminant equal to zero is the necessary and sufficent condition for existence of multiple roots, as it is the existence of common roots for the polynomial and its derivative, which is equivalent to the possibility of expressing the zero polynomial as a combination of the polynomial and its derivative in the described way.
	www.math.niu.edu /~rusin/known-math/95/discriminant (358 words)

	Another new proof of the theorem that every integral rational algebraic function of one variable can be resolved into ...
		The discriminant of the polynomial y is therefore a function of the coefficients l¢, l¢¢, l¢¢¢,...
		symmetrically, that a polynomial of the indeterminates l¢, l¢¢, l¢¢¢,...
		This discriminant of the polynomial z has the same degree: on the other hand the discriminant of the polynomial Z can certainly have lower degree, should some of the coefficients of the higher powers of x vanish.
	www.cs.man.ac.uk /~pt/misc/gauss.html (3655 words)

	Discriminant (Site not responding. Last check: 2007-09-30)
		This function computes the discriminant of a polynomial F (with respect to a given indeterminate X, if the polynomial is multivariate).
		If the polynomial is univariate then there is no need to specify which indeterminate to use.
		The discriminant is defined to be the resultant of F and its derivative with respect to X. Example
	www.math.sunysb.edu /~sorin/online-docs/cocoa/commands/cmd50.html (70 words)

	id:A106273 - OEIS Search Results
		This polynomial is the characteristic polynomial of the Fibonacci and Lucas n-step sequences.
		With that factor removed, the discriminants are prime for odd n=3, 5, 7, 21, 99, 405.
		A086797 (discriminant of the polynomial x^n-x-1), A000045, A000073, A000078, A001591, A001592 (Fibonacci n-step sequences), A000032, A001644, A073817, A074048, A074584 (Lucas n-step sequences), A086937, A106276, A106277, A106278 (number of distinct zeros of these polynomials for n=2, 3, 4, 5).
	www.research.att.com /~njas/sequences/A106273 (206 words)

	Examples (Site not responding. Last check: 2007-09-30)
		The field maximal absolute value of the discriminant allows you to give an upper bound for the field discriminant.
		For example if you enter 3 here, you will only see fields with a discriminant that is a power of 3.
		If you are interested in fields such that the prime factorization of the discriminant contains only primes less than a fixed number you can use the next input form.
	www.mathematik.uni-kassel.de /~klueners/examples.html (273 words)

	discriminant.html
		From this system we eliminate the orginal independent variables to arrive at one polynomial in the continuation parameter t.
		By random choice of complex coefficients in the start system we can ensure that all roots of this discriminant polynomial will miss the interval [0,1].
		As the degree of this "discriminant polynomial" is seven, we can expect seven complex roots:
	www.math.uic.edu /~jan/mcs595f03/discriminant1.html (377 words)

	MuPAD documentation
		polylib::discrim(p, x) returns the discriminant of the polynomial p with respect to the variable x.
		normal is applied to the discriminant before returning it.
		The discriminant of p with respect to the variable x is defined as:
	www.mupad.de /doc/30/de/polylib_discrim.html (81 words)

	[No title]
		What I would like is the complexity of factoring a polynomial p-adically (with degree fixed).
		This is a completely general way to factor integer polynomials if you can factor integers.
		However, I would call it rather ineffective, since its cost grows exponentially with the degree of the polynomial to be factored.
	www.math.niu.edu /~rusin/known-math/95/factor.poly (980 words)

	id:A039744 - OEIS Search Results
		Number of monomials in discriminant of a polynomial of degree n.
		Each monomial in the discriminant of a polynomial of degree n is an integer constant times the product of 2(n-1) of the coefficients, the sum of whose indices (weight) is n(n-1); their number = number of ways n(n-1) can be partitioned into the sum of 2(n-1) integers in the range 0..n.
		Discriminant of cubic K3x^3 + K2x^2 + K1x + K0 is -27K3^2K0^2 + 18K3K2K1K0 - 4K2^3K0 - 4K3K1^3 + K2^2K1^2 which contains 5 monomials.
	www.research.att.com /~njas/sequences/A039744 (136 words)

	Method and system using meta-classes and polynomial discriminant functions for handwriting recognition (US5854855)
		The classifier (32) classifies the handwriting input according to a discriminant function that is based on a polynomial expansion.
		The text is identified according to the discriminant function output.
		Method and apparatus for performing mathematical functions using polynomial approximation and a rectangular aspect ratio multiplier
	www.delphion.com /details?pn10=US05854855 (588 words)

	[No title]
		The resultant of two polynomials over Z[x] and the discriminant of a polynomial over Z[x] can be found.
		The Berlekamp matrix of a squarefree polynomial mod p.
		The Cholesky decomposition of a positive definite matrix is found: A=L*DL, where L is upper unit triangular and D is a diagonal matrix.
	archives.math.utk.edu /software/msdos/linear.algebra/cmat/cmat.readme (985 words)

	Discriminant (Site not responding. Last check: 2007-09-30)
		If a quadratic has real root, its discriminant b2-4ac>=0 Is there any similar condition or method by which you can find whether roots of a cubic equation are real or not?
		Every polynomial equation has a discriminant D, which is the product squared of the diferences of all pairs of roots.
		The original polynomial has three real roots if D is greater or equal zero, and two imaginary roots if D is negative.
	mathcentral.uregina.ca /QQ/database/QQ.09.04/pushkarini1.html (153 words)

	Partial Fractions
		The Fundamental Theorem of Algebra tells us what is possible: Every polynomial can be factored into linear factors (degree 1 polynomials) and irreducible polynomials of degree 2.
		How can you tell whether a degree 2 polynomial is irreducible (over the field of real numbers), or can be factored further into two linear factors?
		Since the discriminant (the expression under the radical) is negative, the polynomial is irreducible!
	www.sosmath.com /algebra/pfrac/pfrac.html (252 words)

	[No title] (Site not responding. Last check: 2007-09-30)
		% EXAMPLE 3.5: % % The example below is a new % attempt at visualizing the discriminant of the % polynomial system % % x^4+ay^2+y=0 % y^4+bx^2+x=0 % % The underlying principal A-determinant % was calculated via maple.
		Theory (looking at % the mixed volume bounds for resultant degrees) % implies that the principal A-determinant should have % degree <= 66.
		This multiple factored % into polynomials of degree 36 and 108, so it would % appear that our desired discriminant is exactly % factor of degree 36, which is presented and % amoebified below...
	www.math.tamu.edu /~rojas/atri.m (197 words)

	Discriminant (Site not responding. Last check: 2007-09-30)
		Let f be a monic polynomial in Q[X] with deg(f) = n different complex zeroes.
		Show that the sign of the discriminant of f is equal to (-1)^s, with 2s the number of non real zeroes of f.
		I know the statement makes sense, because the discriminant is a symmetric polynomial over Q, so it can be written as a polynomial in elementary symmetric polynomials.
	www.physicsforums.com /showthread.php?t=100612 (299 words)

	Weight Discrimination -- Recommendations and Resources (Site not responding. Last check: 2007-09-30)
		Similarly, it is said to have lowest weight λ if λ is a weight and all its other weights are greater than it.
		Other topics related to Weight Discrimination: Weight Distributing
		Categories similar to Weight Discrimination: Weight Of A Gallon Of Water
	www.becomingapediatrician.com /health/168/weight-discrimination.html (1578 words)

	DC MetaData for: Factorization of polynomials modulo small prime powers (Site not responding. Last check: 2007-09-30)
		DC MetaData for: Factorization of polynomials modulo small prime powers
		Abstract: This paper describes an algorith for factoring polynomials
		the polynomial, i.e., the discriminant of the polynomial is zero in the
	math-www.uni-paderborn.de /preprints/metadata/gathen_24.html (57 words)

	Bounds for the Solutions of Some Diophantine Equations in Terms of Discriminants (Site not responding. Last check: 2007-09-30)
		Several effective upper bounds are known for the solutions of Thue equations, Thue-Mahler equations and superelliptic equations.
		One of the basic parameters occuring in these bounds is the height of the polynomial involved in the equation.
		In the present paper it is shown that better (and, in certain important particular cases, best possible) upper bounds can be obtained in terms of the height, if one takes into consideration also the discriminant of the polynomial.
	anziamj.austms.org.au /JAMSA/V51/Part1/Brindza2.html (137 words)

	Homework from February 12 (Site not responding. Last check: 2007-09-30)
		(1) Calculate the discriminant of the curve E: y
		(7) What is the exponent of 3 in the conductor of E? Discriminants
		The list of dates for presentations is on the home page for this class.
	www.willamette.edu /~zizza/Courses/SeniorSeminar/HW4.html (321 words)

	Ba-Bm
		The BAYesian Predictive Discriminant Analysis software implements Bayesian predictive discriminant analysis in Java.
		The aim is to build a model for predicting the value of one discrete variable using other variables.
		This discrimination task is known as classification in the field of machine learning.
	stommel.tamu.edu /~baum/linuxlist/linuxlist/node9.html (11669 words)

	Differential equations associated to algebraic functions and ... (Site not responding. Last check: 2007-09-30)
		The procedure is essentially based on the analytical properties of the algebraic functions and leads to the solution of certain linear differential equations that these functions must satisfy.
		Tschrinhaus transformation, discriminant of a polynomial, entire function, branch and algebraic branch point, generalized hypergeometric functions.
		Source file available in the amsart style of AMS-LaTeX (58.442 bytes).
	www.univie.ac.at /EMIS/journals/RCM/96300102.html (70 words)

	Large number arithmetic in BASIC
		Approximates a real root of integral polynomial f(x) = 0
		Simplify ratios and express decimals as common fractions.
		powering companion matrix M for characteristic polynomial f(x).
	largeint.sourceforge.net (2225 words)

	Factoring over the Complex Numbers, Answer 4
		For which values of a does the polynomial
		The polynomial will have two distinct roots, exactly when the discriminant is positive.
		Please post your question on our S.O.S. Mathematics CyberBoard.
	www.sosmath.com /algebra/factor/fac09a4/fac09a4.html (41 words)

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