Computer Algebra and Polynomials: Applications of Algebra by Jaime Gutierrez, Josef Schicho, Martin Weimann

By Jaime Gutierrez, Josef Schicho, Martin Weimann

Algebra and quantity idea have constantly been counted one of the most pretty mathematical components with deep proofs and stylish effects. notwithstanding, for a very long time they weren't thought of that vital in view of the inability of real-life functions. This has dramatically replaced: these days we discover functions of algebra and quantity concept often in our day-by-day life.

This publication specializes in the speculation and algorithms for polynomials over a variety of coefficient domain names similar to a finite box or ring. The operations on polynomials within the concentration are factorization, composition and decomposition, foundation computation for modules, and so forth. Algorithms for such operations on polynomials have constantly been a imperative curiosity in machine algebra, because it combines formal (the variables) and algebraic or numeric (the coefficients) aspects.

The papers awarded have been chosen from the Workshop on desktop Algebra and Polynomials, which used to be held in Linz on the Johann Radon Institute for Computational and utilized arithmetic (RICAM) in the course of November 25-29, 2013, on the celebration of the distinctive Semester on functions of Algebra and quantity Theory.

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Additional info for Computer Algebra and Polynomials: Applications of Algebra and Number Theory (Lecture Notes in Computer Science)

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To conclude, we follow [20] and examine the structure of the integer points below the line L in more detail, going beyond the closest point (p, q). Define Ta,b to be the triangle with vertices (0, 0), (a, 0) and (a, b). As we can see in Fig. 18, the “staircase” of integer points in Ta,b is irregular: the possible steps as we move from one column to the next are of two different heights, and it is not clear a priori what the underlying pattern is. It turns out, however, that the triangles Ta,b have a very nice recursive structure.

For instance, an explicit computation of (3) in Example 1 shows that this resultant is equal to − 4X22 X02 − X12 − X22 . (6) One may think that the number −4 which appears above is just a random constant, but indeed it is indicating us something very important: if the characteristic of K is 2, then it is easy to verify that (3) does not describe a circle, but the line X2 = 0. What is even worse, (4) is not the parametrization of a curve, as its image is just the point (1 : 1 : 0). To compute the Sylvester Resultant one can use the well-known Sylvester matrix (see [AJ06,CLO07]), whose nonzero entries contain coefficients of the two polynomials X2 u0 (t0 , t1 ) − X0 u2 (t0 , t1 ) and X2 u1 (t0 , t1 ) − X1 u2 (t0 , t1 ), regarded as polynomials in the variables t0 and t1 .

Counting with rational generating functions. J. Symb. Comput. 43(2), 75–91 (2008) 55. : Counting integer points in parametric polytopes using Barvinok’s rational functions. Algorithmica 48(1), 37–66 (2007) 56. : Presburger arithmetic, rational generating functions, and quasipolynomials. , Peleg, D. ) ICALP 2013, Part II. LNCS, vol. 7966, pp. 410–421. Springer, Heidelberg (2013). 0020 57. : Lectures on Polytopes. Graduate Texts in Mathematics. es/personals/dandrea Abstract. In the nineties, several methods for dealing in a more efficient way with the implicitization of rational parametrizations were explored in the Computer Aided Geometric Design Community.

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