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| In algebra, an '''alternating polynomial''' is a [[polynomial]] <math>f(x_1,\dots,x_n)</math> such that if one switches any two of the variables, the polynomial changes sign:
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| :<math>f(x_1,\dots,x_j,\dots,x_i,\dots,x_n) = -f(x_1,\dots,x_i,\dots,x_j,\dots,x_n).</math>
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| Equivalently, if one [[permutes]] the variables, the polynomial changes in value by the [[sign of a permutation|sign of the permutation]]:
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| :<math>f\left(x_{\sigma(1)},\dots,x_{\sigma(n)}\right)= \mathrm{sgn}(\sigma) f(x_1,\dots,x_n).</math>
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| More generally, a polynomial <math>f(x_1,\dots,x_n,y_1,\dots,y_t)</math> is said to be ''alternating in'' <math>x_1,\dots,x_n</math> if it changes sign if one switches any two of the <math>x_i</math>, leaving the <math>y_j</math> fixed.<ref>''Polynomial Identities and Asymptotic Methods,'' [http://books.google.com/books?id=ZLW_Kz_zOP8C&pg=PA12&lpg=PA12&dq=%22alternating+polynomial%22 p. 12]</ref>
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| ==Relation to symmetric polynomials==
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| Products of [[symmetric polynomial|symmetric]] and alternating polynomials (in the same variables <math>x_1,\dots,x_n</math>) behave thus:
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| * the product of two symmetric polynomials is symmetric,
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| * the product of a symmetric polynomial and an alternating polynomial is alternating, and
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| * the product of two alternating polynomials is symmetric.
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| This is exactly the addition table for [[parity (mathematics)|parity]], with "symmetric" corresponding to "even" and "alternating" corresponding to "odd". Thus, the direct sum of the spaces of symmetric and alternating polynomials forms a [[superalgebra]] (a <math>\mathbf{Z}_2</math>-[[graded algebra]]), where the symmetric polynomials are the even part, and the alternating polynomials are the odd part.
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| This grading is unrelated to the grading of polynomials by [[degree of a polynomial|degree]].
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| In particular, alternating polynomials form a [[module (mathematics)|module]] over the algebra of symmetric polynomials (the odd part of a superalgebra is a module over the even part); in fact it is a free module of rank 1, with as generator the [[Vandermonde polynomial]] in ''n'' variables.
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| If the [[Characteristic of a ring|characteristic]] of the coefficient [[ring (mathematics)|ring]] is 2, there is no difference between the two concepts: the alternating polynomials are precisely the symmetric polynomials. | |
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| ==Vandermonde polynomial==
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| {{main|Vandermonde polynomial}}
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| The basic alternating polynomial is the [[Vandermonde polynomial]]:
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| :<math>v_n = \prod_{1\le i<j\le n} (x_j-x_i).</math>
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| This is clearly alternating, as switching two variables changes the sign of one term and does not change the others.<ref>Rather, it only rearranges the other terms: for <math>n=3</math>, switching <math>x_1</math> and <math>x_2</math> changes <math>(x_2-x_1)</math> to <math>(x_1-x_2) = -(x_2-x_1)</math>, and exchanges <math>(x_3-x_1)</math> with <math>(x_3-x_2)</math>, but does not change their sign.</ref>
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| The alternating polynomials are exactly the Vandermonde polynomial times a symmetric polynomial: <math>a = v_n \cdot s</math> where <math>s</math> is symmetric.
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| This is because:
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| * <math>v_n</math> is a factor of every alternating polynomial: <math>(x_j-x_i)</math> is a factor of every alternating polynomial, as if <math>x_i=x_j</math>, the polynomial is zero (since switching them does not change the polynomial, we get
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| :<math>f(x_1,\dots,x_i,\dots,x_j,\dots,x_n) = f(x_1,\dots,x_j,\dots,x_i,\dots,x_n) = -f(x_1,\dots,x_i,\dots,x_j,\dots,x_n),</math>
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| :so <math>(x_j-x_i)</math> is a factor), and thus <math>v_n</math> is a factor.
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| * an alternating polynomial times a symmetric polynomial is an alternating polynomial; thus all multiples of <math>v_n</math> are alternating polynomials
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| Conversely, the ratio of two alternating polynomials is a symmetric function, possibly rational (not necessarily a polynomial), though the ratio of an alternating polynomial over the Vandermonde polynomial is a polynomial.
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| [[Schur polynomial]]s are defined in this way, as an alternating polynomial divided by the Vandermonde polynomial.
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| ===Ring structure===
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| Thus, denoting the ring of symmetric polynomials by Λ<sub>''n''</sub>, the ring of symmetric and alternating polynomials is <math>\Lambda_n[v_n]</math>, or more precisely <math>\Lambda_n[v_n]/\langle v_n^2-\Delta\rangle</math>, where <math>\Delta=v_n^2</math> is a symmetric polynomial, the [[discriminant]].
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| That is, the ring of symmetric and alternating polynomials is a [[quadratic extension]] of the ring of symmetric polynomials, where one has adjoined a square root of the discriminant.
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| Alternatively, it is:
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| :<math>R[e_1,\dots,e_n,v_n]/\langle v_n^2-\Delta\rangle.</math>
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| If 2 is not invertible, the situation is somewhat different, and one must use a different polynomial <math>W_n</math>, and obtains a different relation; see Romagny.
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| ==Representation theory==
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| {{details|Representation theory of the symmetric group}}
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| From the perspective of [[representation theory]], the symmetric and alternating polynomials are subrepresentations of [[Representation theory of the symmetric group|the action of the symmetric group]] on ''n'' letters on the polynomial ring in ''n'' variables. (Formally, the symmetric group acts on ''n'' letters, and thus acts on derived objects, particularly [[free object]]s on ''n'' letters, such as the ring of polynomials.)
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| The symmetric group has two 1-dimensional representations: the trivial representation and the sign representation. The symmetric polynomials are the trivial representation, and the alternating polynomials are the sign representation. Formally, the scalar span of any symmetric (resp., alternating) polynomial is a trivial (resp., sign) representation of the symmetric group, and multiplying the polynomials tensors the representations.
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| In characteristic 2, these are not distinct representations, and the analysis is more complicated.
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| If <math>n>2</math>, there are also other subrepresentations of the action of the symmetric group on the ring of polynomials, as discussed in [[representation theory of the symmetric group]].
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| ==Unstable==
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| Alternating polynomials are an unstable phenomenon (in the language of [[stable homotopy theory]]): the ring of symmetric polynomials in ''n'' variables can be obtained from the ring of symmetric polynomials in arbitrarily many variables by evaluating all variables above <math>x_n</math> to zero: symmetric polynomials are thus ''stable'' or ''compatibly defined.'' However, this is not the case for alternating polynomials, in particular the [[Vandermonde polynomial]].
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| ==Characteristic classes==
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| In [[characteristic classes]], the [[Vandermonde polynomial]] corresponds to the [[Euler class]], and its square (the discriminant) corresponds to the top [[Pontryagin class]]. This is formalized in the [[splitting principle]], which connects characteristic classes to polynomials.
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| From the point of view of [[stable homotopy theory]], the fact that the [[Euler class]] is an unstable class corresponds to the fact that alternating polynomials (and the Vandermonde polynomial in particular) are unstable.
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| ==See also==
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| * [[Symmetric polynomial]]
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| * [[Euler class]]
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| ==Notes==
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| <references/>
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| ==References==
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| * A. Giambruno, Mikhail Zaicev, ''Polynomial Identities and Asymptotic Methods,'' AMS Bookstore, 2005 ISBN 978-0-8218-3829-7, pp. 352
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| * [http://perso.univ-rennes1.fr/matthieu.romagny/notes/FTAF.pdf The fundamental theorem of alternating functions], by Matthieu Romagny, September 15, 2005
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| [[Category:Polynomials]]
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| [[Category:Symmetric functions]]
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Choosing a lawyer can be quite a hard choice. The most crucial section of your decision is what sort of attorney you will need. You require a criminal defense lawyer, if you"re going to trial for a charge. If you are taking a divorce, obviously you want a divorce attorney. Selecting specific representation is obviously advisable as the person you select as your lawyer may have a vast wealth of knowledge on that particular matter, instead of a small amount of knowledge in many different areas. You dont need your own personal injury lawyer to learn such a thing about divorce law right?
So where can you find a reliable attorney? Probably the most convenient way it to check on the web. There really are a few reliable web sites out there for finding an attorney in your city or state. Dig up further on our related paper - Click here: site preview. Generally speaking you desire to search on the basis of the form of illustration you need, followed closely by your state or nearest major city.
What are you searching for in an attorney? You certainly want him to become an honest, warm person. If you have an opinion about jewelry, you will seemingly choose to research about webaddress. Dont waste your time and effort with those who appear to be your not worth theirs. Additionally you need someone with knowledge. This grand truck accident attorney murrieta link has specific provocative suggestions for when to flirt with this viewpoint. Request about experience : just how long he/she has been practicing, what school they graduated from, etc. Most solicitors will joyfully show you their qualifications. If they wait, they probably dont have many credentials and you could need to stay away from that one person.
Over all, when seeking the best attorney you merely gotta use your judgement. The exact same principles apply to choosing a family medical practitioner or even a psychologist; you need to feel comfortable that you"re being cared for. Never be afraid to ask questions, and most importantly keep yourself associated with your situation. If you should be filing for bankruptcy, read up in regards to the exemptions and laws of your particular state..Hales & Associates, Attorneys
41856 Ivy St
Ste #104
Murrieta CA 92562
(951) 489-3320
If you have any queries with regards to the place and how to use family health insurance plans, you can get hold of us at our own website.