EVALUATION PROPERTIES OF SYMMETRIC POLYNOMIALS
نویسندگان
چکیده
منابع مشابه
Evaluation Properties of Symmetric Polynomials
By the fundamental theorem of symmetric polynomials, if P ∈ Q[X1, . . . , Xn] is symmetric, then it can be written P = Q(σ1, . . . , σn), where σ1, . . . , σn are the elementary symmetric polynomials in n variables, and Q is in Q[S1, . . . , Sn]. We investigate the complexity properties of this construction in the straight-line program model, showing that the complexity of evaluation of Q depen...
متن کاملEvaluation techniques and symmetric polynomials
Standard algorithms for dealing with symmetric polynomials are presented using rewriting techniques. This is for instance the case of the ”fundamental theorem of symmetric polynomials”, which states that any symmetric polynomial is a polynomial in the elementary symmetric ones, and whose proof involves rewriting using a suitable elimination order. This kind approach usually spoils useful featur...
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A polynomial invariant under the action of a finite group can be rewritten using generators of the invariant ring. We investigate the complexity aspects of this rewriting process; we show that evaluation techniques enable one to reach a polynomial cost.
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In 1977 G.P. Thomas has shown that the sequence of Schur polynomials associated to a partition can be comfortabely generated from the sequence of variables x = (x1; x2; x3; : : :) by the application of a mixed shift/multiplication operator, which in turn can be easily computed from the set SY T () of standard Young tableaux of shape. We generalise this construction, thereby making possible | fo...
متن کاملSymmetric Polynomials
f(T1, . . . , Tn) = f(Tσ(1), . . . , Tσ(n)) for all σ ∈ Sn. Example 1. The sum T1 + · · ·+ Tn and product T1 · · ·Tn are symmetric, as are the power sums T r 1 + · · ·+ T r n for any r ≥ 1. As a measure of how symmetric a polynomial is, we introduce an action of Sn on F [T1, . . . , Tn]: (σf)(T1, . . . , Tn) = f(Tσ−1(1), . . . , Tσ−1(n)). We need σ−1 rather than σ on the right side so this is a...
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ژورنال
عنوان ژورنال: International Journal of Algebra and Computation
سال: 2006
ISSN: 0218-1967,1793-6500
DOI: 10.1142/s0218196706003128