In mathematics , a polynomialsequence is a sequence of polynomial s indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics , as well as applied mathematics . Examples Some polynomial sequences arise in physics and approximation theory as the solutions of certain ordinary differential equation s Laguerre polynomials Chebyshev polynomials Legendre polynomials Bessel function s Jacobi polynomials Others come from statistics Hermite polynomials Many are studied in algebra and combinatorics Monomial s Rising factorial s Falling factorial s Abel polynomials Bell polynomials Bernoulli polynomials Dickson polynomial s Fibonacci polynomials Lagrange polynomials Lucas polynomials Spread polynomials Touchard polynomials Rook polynomials Classes of polynomial sequences Polynomial sequences of binomial type Orthogonal polynomials Secondary polynomials Sheffer sequence Sturm sequence Generalized Appell polynomials See also Umbral calculus References Aigner, Martin. A course in enumeration , GTM Springer, 2007, ISBN 3 540 39032 4 p21. Roman, Steven The Umbral Calculus , Dover Publications, 2005, ISBN 0 486 44129 3. Williamson, S. Gill Combinatorics for Computer Science , Dover Publications, 2002 p177. DEFAULTSORT PolynomialSequence Category Polynomials Category Sequences and series ar fr Suite de polyn mes it Sequenza polinomiale ... more details
In mathematics , a polynomial is an expression mathematics expression of Finite set finite length constructed ... negative integer Exponentiation exponents . For example, nowrap x sup 2 sup &minus 4 x 7 is a polynomial ... 3 2 . The term polynomial can also be used as an adjective, for quantities that can be expressed as a polynomial of some parameter, as in polynomial time which is used in computational complexity theory . Polynomial comes from the Greek poly , many and medieval Latin binomium , binomial ... www.cnrtl.fr etymologie bin C3 B4me ref ref Etymology of polynomial Compact Oxford English Dictionary .... For example, they are used to form polynomial equations, which encode a wide range of problems, from ... they are used to define polynomial functions, which appear in settings ranging from basic chemistry ... other functions. In advanced mathematics, polynomials are used to construct polynomial ring s, a central concept in abstract algebra and algebraic geometry . Overview A polynomial is either ..., variable should be used only when considering the function defined by the polynomial. In practice ... of a polynomial degree of that variable in that term, the degree of the term is the sum of the degrees of the variables in that term, and the degree of a polynomial is the largest degree of any one ... the degree is 2 1 3. Forming a sum of several terms produces a polynomial. For example, the following is a polynomial math underbrace ,3x 2 begin smallmatrix mathrm term mathrm 1 end smallmatrix ... first, or in ascending powers of x . The polynomial in the example above is written in descending ... nowrap is 5 . The third term is a constant. Since the degree of a non zero polynomial is the largest degree of any one term, this polynomial has degree two. Two terms with the same variables ..., the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied ... 21xy 2x 2y 12x 15y 2 3xy 2 28y 5 ,. math The sum or product of two polynomials is always a polynomial ... more details
of a sequence are polynomial s, then the sequence is a polynomialsequence . If S is endowed ...Other uses In mathematics , a sequence is an ordered list of objects or events . Like a Set mathematics ... possibly infinite is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a Discrete mathematics discrete function mathematics function . For example, C, R, Y is a sequence of letters ..., or Infinite set infinite , such as the sequence of all even and odd numbers even ... notions of sequence, but may be excluded depending on the context. Image Cauchy sequence illustration2.svg right thumb 350px An infinite sequence of real numbers in blue . This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy sequence Cauchy . It is, however, bounded ... of which e.g. , exact sequence are not covered by the notations introduced below. In addition to identifying the elements of a sequence by their position, such as the 3rd element , elements may be given names for convenient referencing. For example a sequence might be written as a sub 1 sub , a sub ... definition of a finite sequence with terms in a set S is a function mathematics function from 1, 2, ..., n to S for some n 0. An infinite sequence in S is a function from 1, 2, ... to S . For example, the sequence of prime numbers 2,3,5,7,11, is the function 1 2 , 2 3 , 3 5 , 4 7 , 5 11 , . A sequence of a finite length n is also called an n tuple n tuple . Finite sequences include the empty sequence ... sequence or two way infinite sequence . An example is the bi infinite sequence of all even integers , 4, 2, 0, 2, 4, 6, 8 . Multiplicative Let A a sequence defined by a function f 1, 2, 3, ... 1, 2, 3, ... , such that a sub i sub f i . The sequence is multiplicative if f xy f x f y for all x , y ... of sequences A subsequence of a given sequence is a sequence formed from the given sequence by deleting ... more details
Infobox musical artist See Wikipedia WikiProject Musicians name The Sequence image caption image size Only for images narrower than 220 pixels background group or band alias origin Columbia, South Carolina Columbia , South Carolina , United States U.S. genre Old school hip hop br Funk years active 1979 1985 label Sugar Hill Records rap Sugar Hill associated acts Spoonie Gee website past members Angie Stone Angie Brown Stone Angie B. br Cheryl Cook Cheryl The Pearl br Gwendolyn Chisolm Blondie The Sequence is a former female old school hip hop trio signed to the Sugar Hill Records rap Sugar Hill label in the early 1980s. The group consisted of Cheryl Cook Cheryl The Pearl , Gwendolyn Chisolm Blondie , and lead singer rapper Angie Stone Angie Brown Stone Angie B. . The group originated from Columbia, South Carolina Columbia , South Carolina as a group of high school cheerleader s. Their most notable single was Funk You Up 1979 , which was the first rap record released by a female group and the second single released by Sugar Hill Records rap Sugar Hill Records . ref name Greenberg1999 Greenberg, Steve Light, Alan ed. 1999 . The VIBE History of Hip Hop . Three Rivers Press. p. 28. ISBN 0609805037 ref Elements of Funk You Up were later used by Dr. Dre for his 1995 single Keep Their Heads Ringin . ref Ego Trip s Book of Rap Lists Book of Rap Lists . 1999. Macmillan Publishers Macmillan ... Hold and later a solo artist. Discography Albums Sugarhill Presents the Sequence 1980 , Sugar Hill Records rap Sugar Hill The Sequence 1982 , Sugar Hill 51 Black Albums The Sequence Party 1983 , Sugar Hill Compilations Funky Sound 1995 , P Vine The Best of the Sequence 1996 , Deep Beats Monster ... The Sequence . Allmusic . External links http www.discogs.com artist Sequence, The Discography DEFAULTSORT Sequence, The Category African American musical groups Category African American rappers Category Female hip hop groups Category Female rappers Category Musical trios Hiphop band stub no The Sequence ... more details
In mathematics, Conway polynomial can refer to the Alexander polynomial Alexander Conway polynomial Alexander Conway polynomial in knot theory the Conway polynomial finite fields disambig ... more details
In mathematics, Carlitz polynomial, named for Leonard Carlitz , may refer to Al Salam Carlitz polynomials Carlitz cyclotomic polynomial Carmichael Carlitz polynomial Dedekind Carlitz polynomial Stieltjes Carlitz polynomial Tricomi Carlitz polynomial mathdab ... more details
Cleanup date March 2009 In mathematics , a sequence math a n n geq 1 math , math a n in mathbb C math obeys a polynomial recurrence of length math ell math if there is a nonzero polynomial math P in mathbb C x 1, dots, x ell math such that math P a n, dots, a n ell 1 0 text for all n geq 1. math ref C. Hillar, L. Levine, Polynomial recurrences and cyclic resultants , Proceedings of the American Mathematical Society , 135 2007 , pages 1607&ndash 1618, http math.mit.edu levine hillarlevineAMSrevised.pdf ref This is at variance with the usual use of the term recurrence relation recurrence which would be more like math a n ell P a n, dots, a n ell 1 text for all n . math Of course this could be transformed into one of the previous form but the converse is not true. The Somos sequence s form examples of integer sequence s generated by a recurrence of this type. Notes reflist algebra stub Category Polynomials Recurrence ... more details
Wikify date September 2011 In mathematics, a biorthogonal polynomial is a polynomial that is orthogonal to several different measures. Biorthogonal polynomials are a generalization of orthogonal polynomials and share many of their properties. There are two different concepts of biorthogonal polynomials in the literature harvtxt Iserles N rsett 1988 introduced the concept of polynomials biorthogonal with respect to a sequence of measures, while Szego introduced the concept of two sequences of polynomials that are birothogonal with respect to each other. Polynomials biorthogonal with respect to a sequence of measures A polynomial p is called biorthogonal with respect to a sequence of measures sub 1 sub , sub 2 sub , ... if math int p x , d mu i x 0 math whenever i deg p . Biorthogonal pairs of sequences Two sequences sub 0 sub , sub 1 sub , ... and sub 0 sub , sub 1 sub , ... of polynomials are called biorthogonal for some measure if math int phi m x psi n x d mu x 0 math whenever m n . The definition of biorthogonal pairs of sequences is in some sense a special case of the definition of biorthogonality with respect to a sequence of measures. More precisely two sequences sub 0 sub , sub 1 sub , ... and sub 0 sub , sub 1 sub , ... of polynomials are biorthogonal for the measure if and only if the sequence sub 0 sub , sub 1 sub , ... is biorthogonal for the sequence of measures sub 0 sub , sub 1 sub , ..., and the sequence sub 0 sub , sub 1 sub , ... is biorthogonal for the sequence of measures sub 0 sub , sub 1 sub ,.... References Citation last1 Iserles first1 Arieh last2 N rsett first2 Syvert Paul title On the theory of biorthogonal polynomials doi 10.2307 2000806 mr 933301 year 1988 journal Transactions of the American Mathematical Society issn 0002 9947 volume 306 issue 2 pages 455 474 Category Orthogonal polynomials ... more details
A polynomial is palindromic, if the sequence of its coefficients are a palindrome . Let math P x sum i 0 n a ix i math be a polynomial of degree n, then P is palindromic if math a i a n i math for i 0...n. Similarly, P is called antipalindromic if math a i a n i math for i 0...n. Examples Some examples of palindromic polynomials are math x 1 2 x 2 2x 1 math math x 1 3 x 3 3x 2 3x 1. math Generally, the expansion of math x 1 n math is palindromic for all n can see this from binomial expansion It also follows that if P is of even degree so has odd number of terms in the polynomial , then it can only be antipalindromic when the middle term is 0, i.e. math a i a i math , where math n 2i math . See also Reciprocal polynomial External links MathPages id home kmath294 kmath294 title The Fundamental Theorem for Palindromic Polynomials algebra stub Category Polynomials Category Palindromes ru ... more details
for the orthogonal polynomials in several variables Hall Littlewood polynomials In mathematics , a Littlewood polynomial is a polynomial all of whose coefficients are 1 or &minus 1. Littlewood s problem asks how large the values of such a polynomial must be on the unit circle in the complex plane . The answer to this would yield information about the autocorrelation of binary sequences. They are named for J. E. Littlewood who studied them in the 1950s. Definition A polynomial math p x sum i 0 n a i x i , math is a Littlewood polynomial if all the math a i pm 1 math . Littlewood s problem asks for constants c sub 1 sub and c sub 2 sub such that there are infinitely many Littlewood polynomials p sub n sub , of increasing degree n satisfying math c 1 sqrt n 1 le p n z le c 2 sqrt n 1 . , math for all math z math on the unit circle. The Rudin Shapiro polynomials provide a sequence satisfying the upper bound with math c 2 sqrt 2 math . No sequence is known as of 2008 which satisfies the lower bound. References cite book author Peter Borwein authorlink Peter Borwein title Computational Excursions in Analysis and Number Theory series CMS Books in Mathematics publisher Springer Verlag year 2002 isbn 0 387 95444 9 pages 2 5,121 132 cite book author J.E. Littlewood authorlink J. E. Littlewood title Some problems in real and complex analysis publisher D.C. Heath year 1968 Category Polynomials Category Conjectures algebra stub ... more details
polynomial is the minimal polynomial field theory minimal polynomial of a primitive element finite ..., a polynomial math F X math with coefficients in GF p Z p Z is a primitive polynomial if it has ... m 2 math is the entire field GF p sup m sup , and moreover, math F X math is the smallest degree polynomial having math alpha math as root. In ring theory , the term primitive polynomial is used for a different purpose, to mean a polynomial over an integral domain R such as the integer s such that no non ... with the ring theory usage. See Gauss s lemma polynomial Gauss s lemma . Properties Because all minimal polynomials are irreducible polynomial irreducible , all primitive polynomials are also irreducible. A primitive polynomial must have a non zero constant term, for otherwise it will be divisible by x . Over the field of two elements, x 1 is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by x 1 . An irreducible polynomial of degree m , F x over GF p for prime p , is a primitive polynomial ... m , where is Euler s totient function . The roots of a primitive polynomial all have order p sup ... of elements of a finite field . If in GF p sup m sup is a root of a primitive polynomial F x ... When these elements are reduced modulo F x they provide the polynomial basis representation of all ... , a primitive polynomials f is a polynomial such that x is a generator of the multiplicative group ... shift register with maximum cycle that is 2 sup lfsr length sup 1 is related with primitive polynomial. For example, given the primitive polynomial x sup 10 sup x sup 3 sup 1, we start with a user ... polynomial of degree m , this process will generate 2 sup m sup &minus 1 pseudo random bits before repeating the same sequence CRC codes The cyclic redundancy check CRC is an error detection code that operates by interpreting the message bitstring as the coefficients of a polynomial over GF ... more details
about the Tutte polynomial of a graph the Tutte polynomial of a matroid Matroid Image Tutte polynomial and chromatic polynomial of the Bull graph.jpg thumb 300px right The polynomial math x 4 x 3 x 2y math is the Tutte polynomial of the Bull graph . The red line shows the intersection with the plane math y 0 math , equivalent to the chromatic polynomial. The Tutte polynomial , also called the dichromate or the Tutte Whitney polynomial , is a polynomial in two variables which plays an important role ... polynomial math T G math comes from the information it contains about math G math . Though originally ... such as the Jones polynomial from knot theory and the partition functions of the Potts model from ... computer science . The Tutte polynomial has several equivalent definitions. It is equivalent to Whitney s rank polynomial , Tutte s own dichromatic polynomial and Fortuin Kasteleyn s random cluster ... For an undirected graph math G V,E math one may define the Tutte polynomial as math T G x ... that math T G math is well defined and a polynomial in math x math and math y math . The same ... a simple change of variables math T G x,y R G x 1,y 1 math . Tutte s dichromatic polynomial math Q ... removed. Then the Tutte polynomial is defined by the recurrence relation math T G T G e T G e ... yet another equivalent definition. The polynomial math Z G q,w sum F subseteq E q k F w F math is equivalent ... 1 right math . Properties The Tutte polynomial factors into connected components If math G math is the union ... polynomial of a planar graph is the flow polynomial of its dual. Examples Isomorphic graphs have the same Tutte polynomial, but the opposite is not true. For example, the Tutte polynomial of every ... j math . For example, the Tutte polynomial of the Petersen graph , math 36 x 120 x 2 180 x 3 170x 4 ... ref R. M. Foster had already observed that the chromatic polynomial is one such function, and Tutte ..., Playing with my W functions I obtained a two variable polynomial from which either the chromatic polynomial ... more details
disambig Minimal polynomial may refer to Minimal polynomial linear algebra Minimal polynomial of a square matrix A , the monic polynomial p x of least degree such that p A 0. Minimal polynomial field theory Minimal polynomial of an algebraic element over a field F , the monic polynomial p x over F of least degree such that p 0. fr Polyn me minimal ru ... more details
In numerical analysis , polynomial interpolation is the interpolation of a given data set by a polynomial given some Point geometry Points in Euclidean geometry points , find a polynomial which goes exactly ... computations. Polynomial interpolation also forms the basis for algorithms in numerical quadrature and numerical ordinary differential equations . Polynomial interpolation is also essential to perform ... , where an interpolation through points on a polynomial which defines the product yields the product ... x sub i sub are the same, one is looking for a polynomial p of degree at most n with the property ... theorem states that such a polynomial p exists and is unique, and can be proved by the Vandermonde matrix , as described below. The theorem states that for n 1 interpolation nodes x sub i sub , polynomial ... n . Constructing the interpolation polynomial Image Interpolation example polynomial.svg thumb ... the interpolation polynomial. Suppose that the interpolation polynomial is in the form math p x a n ... first a Newton polynomial Newton interpolation of the polynomial and then converting it to the monomial form above. Uniqueness of the interpolating polynomial Proof 1 Suppose we interpolate through n 1 data points with an at most n degree polynomial p x we need at least n 1 datapoints or else the polynomial cannot be fully solved for . Suppose also another polynomial exists ... math r x p x q x math . We know, r x is a polynomial r x has degree at most n , since math p x math ... x is an n degree polynomial or less It has one root too many. Formally, if math r x math is any non zero polynomial, it must be writable as math r x x x 0 x x 1 cdots x x n math . By distributivity the n ... q x math So math q x math which could be any polynomial, so long as it interpolates the points is identical ... what method we use to do our interpolation direct, Spline mathematics spline , Lagrange polynomial lagrange etc., assuming we can do all our calculations perfectly we will always get the same polynomial ... more details
Auxiliary polynomial is a term in mathematics which may refer to The auxiliary function argument in transcendence theory The characteristic polynomial of a recurrence relation mathdab ... more details
In algebra, a multilinear polynomial is a polynomial that is linear in each of its variables. In other words, no variable occurs to a power of 2 or higher or alternatively, each monomial is a constant times a product of distinct variables. They are important in the study of polynomial identity testing , because if a multilinear polynomial is zero on a set of vectors that Linear span span the space, it will be zero everywhere. The Degree of a polynomial degree of a multilinear polynomial is the maximum number of distinct variables occurring in any monomial. ref A. Giambruno, Mikhail Zaicev. Polynomial Identities and Asymptotic Methods. AMS Bookstore, 2005 ISBN 978 0 82183829 7. Section 1.3. ref References references Category Polynomials ... more details
In the mathematics mathematical field of knot theory , a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot mathematics knot . History The first knot polynomial, the Alexander polynomial , was introduced by J. W. Alexander in 1923, but other knot polynomials were not found until almost 60 years later. In the 1960s, John Horton Conway John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander&ndash Conway polynomial . The significance of this skein relation was not realized until the early 1980s, when Vaughan Jones discovered the Jones polynomial . This led to the discovery of more knot polynomials, such as the so called HOMFLY polynomial . Soon after Jones discovery, Louis Kauffman noticed the Jones polynomial could be computed by means of a state sum model , which involved the bracket polynomial , an invariant of framing knot framed knots. This opened up avenues of research linking knot theory and statistical mechanics . In the late 1980s, two related breakthroughs were made. Edward Witten demonstrated that the Jones polynomial, and similar Jones type invariants, had an interpretation in Chern&ndash Simons theory . Victor Anatolyevich Vasilyev Viktor Vassiliev and Mikhail Goussarov started the theory of finite type invariant s of knots. The coefficients of the previously named polynomials are known to be of finite type after perhaps a suitable change of variables . In recent years, the Alexander polynomial has been shown to be related to Floer homology . The graded Euler characteristic of the Heegaard Floer homology knot Floer homology of Ozsv th and Szab is the Alexander polynomial. References Colin Adams, The Knot Book , American ... knot polynomials Alexander polynomial Bracket polynomial HOMFLY polynomial Jones polynomial Kauffman polynomial Related topics skein relationship for a formal definition of the Alexander polynomial ... more details
In mathematics , in the realm of abstract algebra , a radical polynomial is a multivariate polynomial over a field that can be expressed as a polynomial in the sum of squares of the variables. That is, if math k x 1, x 2, ldots, x n math is a polynomial ring , the ring of radical polynomials is the subring generated by the polynomial math sum i 1 n x i 2. math Radical polynomials are characterized as precisely those polynomials that are invariant mathematics invariant under the action of the orthogonal group . The ring of radical polynomials is a graded algebra graded subalgebra of the ring of all polynomials. The standard separation of variables theorem asserts that every polynomial can be expressed as a finite sum of terms, each term being a product of a radical polynomial and a harmonic polynomial . This is equivalent to the statement that the ring of all polynomials is a free module over the ring of radical polynomials. References unreferenced date June 2008 Category Abstract algebra Category Polynomials Category Invariant theory Abstract algebra stub ... more details
Merge to Chebyshev polynomials discuss Talk Chebyshev polynomials Dickson polynomial date September 2011 In mathematics , the Dickson polynomials , denoted D sub n sub x , , form a polynomialsequence studied by harvs txt yes authorlink Leonard Eugene Dickson first L. E. last Dickson year 1897 . Over the complex numbers, Dickson polynomials are essentially equivalent to Chebyshev polynomial s with a change of variable, and in fact Dickson polynomials are sometimes called Chebyshev polynomials. Dickson polynomials are mainly studied over finite fields, when they are not equivalent to Chebyshev polynomials. One of the main reasons for interest in them is that for fixed , they give many examples of permutation polynomials polynomials acting as permutations of finite fields. Definition D sub 0 sub x , 2, and for n 0 Dickson polynomials of the first kind are given by math D n x, alpha sum p 0 lfloor n 2 rfloor frac n n p binom n p p alpha p x n 2p . math The first few Dickson polynomials are math D 0 x, alpha 2 , math math D 1 x, alpha x , math math D 2 x, alpha x 2 2 alpha , math math ... x, alpha . , math The Dickson polynomial D sub n sub y is a solution of the ordinary differential equation math x 2 4 alpha y xy n 2y 0 , math and the Dickson polynomial E sub n sub y is a solution of the differential ... Dickson polynomials over the complex numbers are related to Chebyshev polynomial s T sub n sub ..., the Dickson polynomial D sub n sub x , a can be defined over rings in which a is not a square, and over ... polynomial. The Dickson polynomials with parameter 1 or 1 are related to the Fibonacci polynomials ... D n x,0 x n , . math Permutation polynomials and Dickson polynomials A permutation polynomial for a given ... polynomial D sub n sub x , considered as a function of x with fixed is a permutation polynomial ... proved that any integral polynomial that is a permutation polynomial for infinitely many prime fields ... permutation polynomial over the finite field F sub q sub whose degree is simultaneously coprime to q ... more details
In mathematics , in abstract algebra , a multivariate polynomial over a field whose Laplacian is zero is termed a harmonic polynomial . The harmonic polynomials form a vector space vector subspace of the vector space of polynomials over the field. In fact, they form a graded algebra graded subspace . The Laplacian is the sum of second partials with respect to all the variables, and is an invariant mathematics invariant differential operator under the action of the orthogonal group viz the Group mathematics group of rotations. The standard separation of variables theorem states that every multivariate polynomial over a field can be decomposed as a finite sum of products of a radical polynomial and a harmonic polynomial. This is equivalent to the statement that the polynomial ring is a free module over the ring of radical polynomials. References Lie Group Representations of Polynomial Rings by Bertram Kostant published in the American Journal of Mathematics Vol 85 No 3 July 1963 algebra stub Category Abstract algebra Category Polynomials ru uk ... more details
In knot theory , the Kauffman polynomial is a 2 variable knot polynomial due to Louis Kauffman . It is initially defined on a link knot theory link diagram as math F K a,z a w K L K , math where math w K math is the writhe of the link diagram and math L K math is a polynomial in a and z defined on link diagrams by the following properties math L O 1 math O is the unknot math L s r aL s , qquad L s ell a 1 L s . math L is unchanged under type II and III Reidemeister move s Here math s math is a strand and math s r math resp. math s ell math is the same strand with a right handed resp. left handed curl added using a type I Reidemeister move . Additionally L must satisfy Kauffman s skein relation Image Kauffman poly.png 400px The pictures represent the L polynomial of the diagrams which differ inside a disc as shown but are identical outside. Kauffman showed that L exists and is a regular isotopy invariant of unoriented links. It follows easily that F is an ambient isotopy invariant of oriented links. The Jones polynomial is a special case of the Kauffman polynomial, as the L polynomial specializes to the bracket polynomial . The Kauffman polynomial is related to Chern Simons theory Chern Simons gauge theories for SO N in the same way that the HOMFLY polynomial is related to Chern Simons gauge theories for SU N see Witten s article Quantum field theory and the Jones polynomial , in Commun. Math. Phys. References Louis Kauffman , On Knots , 1987 , ISBN 0 691 08435 1 External links http eom.springer.de k k120040.htm Springer EoM entry for Kauffman polynomial http katlas.math.toronto.edu wiki The Kauffman Polynomial Knot Atlas entry for Kauffman polynomial Category Knot theory Category Polynomials knottheory stub ... more details
In coding theory , the weight enumerator polynomial of a binary linear code specifies the number of words of each possible Hamming weight . Let math C subset mathbb F 2 n math be a binary linear code length math n math . The weight distribution is the sequence of numbers math A t c in C mid w c t math giving the number of codeword s c in C having weight t as t ranges from 0 to n . The weight enumerator is the bivariate polynomial math W C x,y sum w 0 n A w x w y n w . math Basic properties math W C 0,1 A 0 1 math math W C 1,1 sum w 0 n A w C math math W C 1,0 A n 1 mbox iff 1, ldots,1 in C mbox and 0 mbox otherwise. math math W C 1, 1 sum w 0 n A w 1 n w A n 1 1 A n 1 ldots 1 n 1 A 1 1 n A 0 math MacWilliams identity Denote the dual code of math C subset mathbb F 2 n math by math C perp x in mathbb F 2 n , mid , langle x,c rangle 0 mbox forall c in C math where math , math denotes the vector dot product and which is taken over math mathbb F 2 math . The MacWilliams identity states that math W C perp x,y frac 1 mid C mid W C y x,y x . math The identity is named after Jessie MacWilliams . Distance enumerator The distance distribution or inner distribution of a code C of size M and length n is the sequence of numbers math A i frac 1 M left lbrace c 1,c 2 in C times C mid d c 1,c 2 i right rbrace math where i ranges from 0 to n . The distance enumerator polynomial is math A C x,y sum i 0 n A i x i y n i math and when C is linear this is equal to the weight enumerator. The outer distribution of C is the 2 sup n sup by n 1 matrix B with rows indexed by elements of GF 2 sup n sup and columns indexed by integers 0... n , and entries math B x,i left lbrace c in C mid d c,x i right rbrace . math The sum of the rows of B is M times the inner distribution vector A sub 0 sub ,..., A sub n sub . A code C is regular if the rows of B corresponding to the codewords of C are all equal. References cite book last Hill first Raymond title A first course in coding theory publisher Oxford ... more details
distinguish matrix polynomial In mathematics , a polynomial matrix or sometimes matrix polynomial is a matrix mathematics matrix whose elements are univariate or multivariate polynomial s. A matrix is a matrix whose elements are polynomials in . A univariate polynomial matrix P of degree p is defined as math P sum n 0 p A n x n A 0 A 1 x A 2 x 2 cdots A p x p math where math A i math denotes a matrix of constant coefficients, and math A p math is non zero. Thus a polynomial matrix is the matrix equivalent of a polynomial, with each element of the matrix satisfying the definition of a polynomial of degree p . An example 3 3 polynomial matrix, degree 2 math P begin pmatrix 1 & x 2 & x 0 & 2x & 2 3x 2 & x 2 1 & 0 end pmatrix begin pmatrix 1 & 0 & 0 0 & 0 & 2 2 & 1 & 0 end pmatrix begin pmatrix 0 & 0 & 1 0 & 2 & 0 3 & 0 & 0 end pmatrix x begin pmatrix 0 & 1 & 0 0 & 0 & 0 0 & 1 & 0 end pmatrix x 2. math We can express this by saying that for a ring R , the rings math M n R X math and math M n R X math are Ring homomorphism isomorphic . Properties A polynomial matrix over a field mathematics field with determinant equal to a non zero element of that field is called unimodular matrix unimodular , and has an inverse that is also a polynomial matrix. Note that the only scalar unimodular polynomials are polynomials of degree 0 nonzero constants, because an inverse of an arbitrary polynomial of higher degree is a rational function. The roots of a polynomial matrix over the complex numbers are the points in the complex plane where the matrix loses rank linear algebra rank . Note that polynomial matrices are not to be confused with monomial matrix monomial matrices , which are simply ... let A be a polynomial matrix, then the matrix I A is the characteristic matrix of the matrix A . Its determinant, I A is the characteristic polynomial of the matrix A . References E.V.Krishnamurthy, Error free Polynomial Matrix computations, Springer Verlag, New York, 1985 Category Matrices Category ... more details
citations missing date February 2011 In mathematics , and in particular in the field of algebra , a polynomial expression in one or more given entities E sub 1 sub , E sub 2 sub , ..., is any meaningful expression constructed from copies of those entities together with constant mathematics constant s, using the operations of addition and multiplication. For each entity E , multiple copies can be used, and it is customary to write the product E E ... E of some number n of identical copies of E as E sup n sup thus the operation of raising to a constant natural number power may also be used as abbreviation in a polynomial expression. Similarly, subtraction X Y may be used to abbreviate X 1 Y . The entities used may be of various natures. They are usually not explicitly given values, since then the polynomial expression can just be evaluated to another such value. Often they are symbols such as x , &lambda or X , which according to the context may stand for an unknown ... the polynomial expression is just a polynomial . It is however also possible to form polynomial ... of polynomial expressions. The entities may be themselves expressions, not necessarily polynomial ones ... as a polynomial expression in the entity cos x , as in cos 3 x 4 cos x sup 3 sup &minus 3 cos x . Here it would be incorrect to call the right hand side a polynomial ... a certain polynomial expression in A to the null matrix. The entries may be somewhat unknown quantities without being completely free variables. For instance, for any Polynomial Classifications monic polynomial of degree n that has n roots, Vi te s formulas express its coefficients as symmetric polynomial ... independently of the choice of such a polynomial therefore the n roots are not known values as they would be if the polynomial were fixed , but they are not variables or indeterminates either. See also Polynomial References reflist Category Abstract algebra Category Polynomials ... more details
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