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Finite Fields
Finite Fields

... • the integers (Z, +, ∗) form an integral domain but not a field; • the rationals (Q, +, ∗), reals (R, +, ∗) and complex numbers (C, +, ∗) form fields; • the set of 2 × 2 matrices with real entries forms a non-commutative ring with identity w.r.t. matrix addition and multiplication. • the group Zn w ...
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... the inverse of the generic n × n matrix X requires n nested inversions, namely has height n. Our lower bound on formula size of matrix inverse is obtained by showing that a formula of size s can compute a function of height at most logarithmic in s. This is obtained via a general balancing procedure ...
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... 1. Clearly, it is enough to check it for f(x) = xk , since every polynomial is a linear combination of these, and if x − a divides each of the summands, it divides the whole sum too. But xk − ak = (x − a)(xk−1 + xk−2 a + . . . + xak−2 + ak−1 ). The statement about the roots is clear: f(x) = q(x)(x − ...
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... Arithmetic circuit complexity studies the computation of polynomials and rational functions using the basic operations addition, multiplication, and division. It is chiefly interested in commutative polynomials or rational functions, defined over a set of multiplicatively commuting variables (see th ...
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... Parts of the composition Ie conjugate to a composition I can be read from the diagram of the composition I from left to right and from bottom to top. A partition is a composition with weakly decreasing parts, i.e. λ = (λ1 , . . . , λn ) with λ1 ≥ λ2 ≥ . . . ≥ λn The number of times an integer i occu ...
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Gröbner basis

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field K[x1, ..,xn]. A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps.Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, andGaussian elimination for linear systems.Gröbner bases were introduced in 1965, together with an algorithm to compute them (Buchberger's algorithm), by Bruno Buchberger in his Ph.D. thesis. He named them after his advisor Wolfgang Gröbner. In 2007, Buchberger received the Association for Computing Machinery's Paris Kanellakis Theory and Practice Award for this work.However, the Russian mathematician N. M. Gjunter had introduced a similar notion in 1913, published in various Russian mathematical journals. These papers were largely ignored by the mathematical community until their rediscovery in 1987 by Bodo Renschuch et al. An analogous concept for local rings was developed independently by Heisuke Hironaka in 1964, who named them standard bases.The theory of Gröbner bases has been extended by many authors in various directions. It has been generalized to other structures such as polynomials over principal ideal rings or polynomial rings, and also some classes of non-commutative rings and algebras, like Ore algebras.
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