Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

Unbiased Stereology. Three‐Dimensional Measurement in Microscopy.

1 S-polynomials eliminating the leading term Buchberger's criterion and algorithm 2 Wavelet Design construct wavelet filters 3 Proof of the Buchberger Criterion two lemmas proof of the Buchberger criterion termination and elimination

Gröbner Bases와 응용

In this paper we consider a broad family of toroidal networks, denoted as Gaussian networks, which include many previously proposed and used topologies. We will define such networks by means of the Gaussian integers, the subset of the complex numbers with integer real and imaginary parts. Nodes in Gaussian networks are labeled by Gaussian integers, which confer these topologies an algebraic structure based on quotient rings of the Gaussian integers. In this sense, Gaussian integers reveal themselves as the appropriate tool for analyzing and exploiting any type of toroidal network. Using this algebraic approach, we can characterize the main distance-related properties of Gaussian networks, providing closed expressions for their diameter and average distance. In addition, we solve some important applications, like unicast and broadcast packet routing or the perfect placement of resources over these networks.

Modeling Toroidal Networks with the Gaussian Integers

We obtain some theoretical and experimental results concerning various properties (the number of fixed points, image distribution, cycle lengths) of the dynamical system naturally associated with Fermat quotients acting on the set $\{0,\dots,p-1\}$. In particular, we improve the lower bound of Vandiver [Bull. Amer. Math. Soc., 22 (1915), pp. 61-67] on the image size of Fermat quotients on the above set (from $p^{1/2}-1$ to $(1+o(1))p(\log p)^{-2}$). We also consider pseudorandom properties of Fermat quotients such as uniform distribution and linear complexity.

Pseudorandomness and Dynamics of Fermat Quotients

We prove a bound on sums of products of multiplicative characters of shifted Fermat quotients modulo p. From this bound we derive results on the pseudorandomness of sequences of modular discrete logarithms of Fermat quotients modulo p: bounds on the well-distribution measure, the correlation measure of order l, and the linear complexity.

Multiplicative character sums of Fermat quotients and pseudorandom sequences

https://personales.unican.es/gomezd/Articles/Ibeas/TheoreticalComputerScience.pdf

Optimal routing in double loop networks

We show that the multiplicity of a prime p as a factor of the resultant of two polynomials with integer coefficients is at least the degree of their greatest common divisor modulo p. This answers an open question by Konyagin and Shparlinski.

/pdf/common-factors-of-resultants-modulo-p-1q4yjifphw.pdf

COMMON FACTORS OF RESULTANTS MODULO p

https://personales.unican.es/gomezd/Articles/Nicolas/critical-orbits.pdf

An estimate on the number of stable quadratic polynomials

Let p be a prime and let a and c be integers modulo p. The quadratic congruential generator (QCG) is a sequence (vn) of pseudorandom numbers defined by the relation $v_{n+1}\equiv av^{2}_{n}+c mod p$. We show that if sufficiently many of the most significant bits of several consecutive values vn of the QCG are given, one can recover in polynomial time the initial value v0 (even in the case where the coefficient c is unknown), provided that the initial value v0 does not lie in a certain small subset of exceptional values.

/pdf/cryptanalysis-of-the-quadratic-generator-48049jnxp0.pdf

Domingo Gomez

Papers

Multiplicative character sums of Fermat quotients and pseudorandom sequences

Optimal routing in double loop networks

COMMON FACTORS OF RESULTANTS MODULO p

An estimate on the number of stable quadratic polynomials

Cryptanalysis of the quadratic generator