Affine projections of symmetric polynomials
نویسندگان
چکیده
منابع مشابه
Affine Projections of Symmetric Polynomials
In this paper we introduce a new model for computing polynomials a depth 2 circuit with a symmetric gate at the top and plus gates at the bottom, i.e the circuit computes a symmetric function in linear functions Sd m(`1; `2; :::; `m) (Sd m is the d’th elementary symmetric polynomial in m variables, and the `i’s are linear functions). We refer to this model as the symmetric model. This new model...
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An m-variate polynomial f is said to be an affine projection of some n-variate polynomial g if there exists an n×m matrix A and an n-dimensional vector b such that f(x) = g(Ax + b). In other words, if f can be obtained by replacing each variable of g by an affine combination of the variables occurring in f , then it is said to be an affine projection of g. Given f and g can we determine whether...
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The power symmetric polynomial on n variables of degree d is defined as pd(x1, . . . , xn) = x d 1 + · · · + xn. We study polynomials that are expressible as a sum of powers of homogenous linear projections of power symmetric polynomials. These form a subclass of polynomials computed by depth five circuits with summation and powering gates (i.e., ∑∧∑∧∑ circuits). We show 2Ω(n) size lower bounds...
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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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Bivariate Gončarov polynomials are a basis of the solutions of the bivariate Gončarov Interpolation Problem in numerical analysis. A sequence of bivariate Gončarov polynomials is determined by a set of nodes Z = {(xi,j, yi,j) ∈ R2} and is an affine sequence if Z is an affine transformation of the lattice grid N2, i.e., (xi,j, yi,j) = A(i, j)T + (c1, c2) for some 2 × 2 matrix A and constants c1,...
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ژورنال
عنوان ژورنال: Journal of Computer and System Sciences
سال: 2002
ISSN: 0022-0000
DOI: 10.1016/s0022-0000(02)00021-1