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Algebraic Geometry
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Let L/K be a field extension and α∈L. If there is a polynomial f(x)∈K[x] such that f(α)=0, then α is called an algebraic element over K. An algebraic extension is a field extension L/K such that every element of L is algebraic over K.
A linear algebra method to decompose forms whose length is lower than the number of variables into weighted sum of squares
Published in International Journal of Control, 2019
Laura Menini, Corrado Possieri, Antonio Tornambè
Let n = 3, d = 2 and let be the field of all the rational functions in π with coefficients in the algebraic extension of , that is the smallest algebraic extension of that contains . Thus, consider the form By using Algorithm 1 with input p, one obtains the following polynomials By employing Algorithm 2 with inputs p and g1, g2, one obtains that the polynomial p can be rewritten as Thus, by applying Algorithm 3 to such a quadratic representation, one obtains that the form p can be rewritten as the following wSOS+
SPECTRA – a Maple library for solving linear matrix inequalities in exact arithmetic
Published in Optimization Methods and Software, 2019
Didier Henrion, Simone Naldi, Mohab Safey El Din
For example, for the point the polynomial q in the rational parametrization (1) is Recall that the algebraic degree of a point in is the degree of the minimal algebraic extension of the ground field (here the rational numbers) required to represent . The algebraic degree depends on the size of the pencil A but also on the rank r of . With and generic data, the algebraic degree is 12, cf. [15, Table 2], which indeed coincides with the degree of the exact representation of computed by spectra.
Algebraic differentiators through orthogonal polynomials series expansions
Published in International Journal of Control, 2018
All through this paper: denotes a field of characteristic zero (e.g. , or );for a set , the algebraic extension of by is denoted by .