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

DORIN GHISA Riemann Zeta function is one of the most studied transcendental functions, having in view its many applications in number theory,algebra, complex analysis, statistics, as well as in physics. Another reason why this function has drawn so much attention is the celebrated Riemann conjecture regarding its non trivial zeros, which resisted proof or disproof until now. There has been lately a lot of progress in the sense of proving its truthfulness.

/pdf/the-riemann-zeta-function-3l59kypwcz.pdf

The Riemann zeta function

The first goal of algebraic number theory is the generalization of the theorem on the unique representation of natural numbers as products of prime numbers to algebraic numbers. Gauss considered the ring $ \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] $ of all numbers of the form $ a + \sqrt {{ - 1b}} $ with $ a,\;b \in \mathbb{Z} $ and showed that $ \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] $ is a ring with unique factorization in prime elements (see §2.1). He introduced these numbers for the development of his theory of biquadratic residues. Another motivation for the study of the arithmetic of algebraic numbers comes from the theory of Tiophantine equations. For example, the quadratic form $ f({x_1},\;{x_2}) = x_1^2 - Dx_2^2 $ with $ D \in \mathbb{Z} $, $ \sqrt {{D\; \notin \;\mathbb{Z}}} $ can be written in the form $ \left( {{x_1} - \sqrt {{D{x_2}}} } \right)\left( {{x_1} + \sqrt {{D{x_2}}} } \right) $. Hence the question about the representation of integers by f(a 1, a 2) with $ {a_1},\;{a_2} \in \mathbb{Z} $ can be reformulated as a question of factorization of algebraic numbers of the form $ {a_1} + \sqrt {{D{a_2}}} $. These numbers form a module in the field $ \mathbb{Q}(\sqrt {D} ) $.

Basic Number Theory

We give a new explicit construction of n×N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ϵ>0, large N, and any n satisfying N1−ϵ≤n≤N, we construct RIP matrices of order k≥n1/2+ϵ and constant δ=n−ϵ. This overcomes the natural barrier k=O(n1/2) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose kth moments are uniformly small for 1≤k≤N (Turan's power sum problem), which improves upon known explicit constructions when (logN)1+o(1)≤n≤(logN)4+o(1). This latter construction produces elementary explicit examples of n×N matrices that satisfy the RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (logN)1+o(1)≤n≤(logN)5/2+o(1).

/pdf/explicit-constructions-of-rip-matrices-and-related-problems-ku4kj61bj1.pdf

Explicit constructions of RIP matrices and related problems

Let A be a thin set in Z n , and A N the intersection of A with the ball of diameter N centred at the origin. When n = 1 we have seen in §9.7 (as a consequence of Siegel’s theorem) that |A N | = O(N 1/2) when N → ∞. To prove a similar result when n ≥ 2 one needs a different method, based on the large sieve inequality (cf. [Co]).

The Large Sieve

Let Fp be the field of a prime order p and F*p be its multiplicative subgroup In this paper we obtain a variant of sum-product estimates which in particular implies the bound for any subset A ⊂ Fp with 1 p1/4, then Finally we show that if g is a generator of F*p then for any M < p the number of solutions of the equation is less than  This implies that if p1/2 < M < p, then

On a variant of sum-product estimates and explicit exponential sum bounds in prime fields

An Explicit Sum-Product Estimate in p

Let $\mathbb{F}_p$ be the field of a prime order $p.$ It is known that for any integer $N\in [1,p]$ one can construct a subset $A\subset\mathbb{F}_p$ with $|A|= N$ such that $$ \max\{|A+A|, |AA|\}\ll p^{1/2}|A|^{1/2}. $$ In the present paper we prove that if $A\subset \mathbb{F}_p$ with $|A|>p^{2/3},$ then $$ \max\{|A+A|, |AA|\}\gg p^{1/2}|A|^{1/2}. $$

The sum-product estimate for large subsets of prime fields

Let Fp be the field of prime order p. It is known that for any integer N ∈ [1,p] one can construct a subset A C Fp with |A| = N such that max{|A + A|, |AA|} «p 1/2 |A| 1/2 . One of the results of the present paper implies that if A C Fp with |A| > p 2/3 , then max{|A + A|, |AA|}» p 1/2 |A| 1/2 .

https://www.ams.org/proc/2008-136-08/S0002-9939-08-09386-6/S0002-9939-08-09386-6.pdf

Let p be a large prime number, K, L, M, λ be integers with 1 ≤ M ≤ p and gcd(λ, p) = 1. The aim of our paper is to obtain sharp upper bound estimates for the number I2(M; K, L) of solutions of the congruence 

$$xy \equiv \lambda \quad ({\rm mod} p), \quad K+ 1 \leq x \leq K +M,\quad L+ 1 \leq y \leq L +M,$$

and for the number I3(M;L) of solutions of the congruence 

$$xyz \equiv \lambda \quad ({\rm mod} p), \quad L+ 1 \leq x, y, z \leq L +M .$$

Using the idea of Heath-Brown from [H], we obtain a bound for I2(M; K, L), which improves several recent results of Chan and Shparlinski [CS]. For instance, we prove that if M < p1/4, then I2(M; K, L) ≤ Mo(1).

/pdf/concentration-of-points-on-two-and-three-dimensional-modular-5arr6lbyy3.pdf

Moubariz Z. Garaev

Papers

On a variant of sum-product estimates and explicit exponential sum bounds in prime fields

An Explicit Sum-Product Estimate in p

The sum-product estimate for large subsets of prime fields

The sum-product estimate for large subsets of prime fields

Concentration of Points on Two and Three Dimensional Modular Hyperbolas and Applications