Convex bodies: the brunn–minkowski theory

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

Steganography, the art of hiding of information in apparently innocuous objects or images, is a field with a rich heritage, and an area of rapid current development. This clear, self-contained guide shows you how to understand the building blocks of covert communication in digital media files and how to apply the techniques in practice, including those of steganalysis, the detection of steganography. Assuming only a basic knowledge in calculus and statistics, the book blends the various strands of steganography, including information theory, coding, signal estimation and detection, and statistical signal processing. Experiments on real media files demonstrate the performance of the techniques in real life, and most techniques are supplied with pseudo-code, making it easy to implement the algorithms. The book is ideal for students taking courses on steganography and information hiding, and is also a useful reference for engineers and practitioners working in media security and information assurance.

http://www.gbv.de/dms/ilmenau/toc/608077828.PDF

Steganography in Digital Media: Principles, Algorithms, and Applications

Poised to become the leading reference in the field, the Handbook of Finite Fields is exclusively devoted to the theory and applications of finite fields. More than 80 international contributors compile state-of-the-art research in this definitive handbook. Edited by two renowned researchers, the book uses a uniform style and format throughout and each chapter is self contained and peer reviewed. The first part of the book traces the history of finite fields through the eighteenth and nineteenth centuries. The second part presents theoretical properties of finite fields, covering polynomials, special functions, sequences, algorithms, curves, and related computational aspects. The final part describes various mathematical and practical applications of finite fields in combinatorics, algebraic coding theory, cryptographic systems, biology, quantum information theory, engineering, and other areas. The book provides a comprehensive index and easy access to over 3,000 references, enabling you to quickly locate up-to-date facts and results regarding finite fields.

Handbook of Finite Fields

Annals of mathematics studies

We present a new approach to the problem of computing the zeta function of a hypersurface over a finite field. For a hypersurface defined by a polynomial of degree $d$ in $n$ variables over the field of $q$ elements, one desires an algorithm whose running time is a polynomial function of $d^n \log(q)$. (Here we assume $d \geq 2$, for otherwise the problem is easy.) The case $n = 1$ is related to univariate polynomial factorisation and is comparatively straightforward. When $n = 2$ one is counting points on curves, and the method of Schoof and Pila yields a complexity of $\log(q)^{C_d}$, where the function $C_d$ depends exponentially on $d$. For arbitrary $n$, the theorem of the author and Wan gives a complexity which is a polynomial function of $(pd^n \log(q))^n$, where $p$ is the characteristic of the field. A complexity estimate of this form can also be achieved for smooth hypersurfaces using the method of Kedlaya, although this has only been worked out in full for curves. The new approach we present should yield a complexity which is a small polynomial function of $pd^n\log(q)$. In this paper, we work this out in full for Artin?Schreier hypersurfaces defined by equations of the form $Z^p - Z = f$, where the polynomial $f$ has a diagonal leading form. The method utilises a relative $p$-adic cohomology theory for families of hypersurfaces, due in essence to Dwork. As a corollary of our main theorem, we obtain the following curious result. Let $f$ be a multivariate polynomial with integer coefficients whose leading form is diagonal. There exists an explicit deterministic algorithm which takes as input a prime $p$, outputs the number of solutions to the congruence equation $f = 0 \bmod{p}$, and runs in ${\mathcal O}(p^{2 + \varepsilon})$ bit operations, for any $\varepsilon > 0$. This improves upon the elementary estimate of ${\mathcal O}(p^{n-1 + \varepsilon})$ bit operations, where $n$ is the number of variables, which can be achieved using Berlekamp's root counting algorithm.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0024611503014461

Deformation Theory and The Computation of Zeta Functions

Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral polygons is NP-complete then present a pseudo-polynomial-time algorithm for decomposing polygons. For higher-dimensional polytopes, we give a heuristic algorithm which is based upon projections and uses randomization. Applications of our algorithms include absolute irreducibility testing and factorization of polynomials via their Newton polytopes.

/pdf/decomposition-of-polytopes-and-polynomials-v95nxac73j.pdf

Decomposition of Polytopes and Polynomials

Let be odd two-dimensional Artin representations for which is self-dual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms of respective weights two, one, and one, giving rise to via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain -adic iterated integrals attached to the triple -adic avatars of the leading term of the Hasse–Weil–Artin when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on —referred to as Stark points—which are defined over the number field cut out by . This formula can be viewed as an elliptic curve analogue of Stark’s conjecture on units attached to weight-one forms. It is proved when are binary theta series attached to a common imaginary quadratic field in which splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing -adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell–Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark–Heegner points attached to Shintani-type cycles on with Galois group a central extension of the dihedral group .

/pdf/stark-points-and-adic-iterated-integrals-attached-to-modular-23qjofdef5.pdf

Stark points and -adic iterated integrals attached to modular forms of weight one

We present a deterministic polynomial time algorithm for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic. One consequence of this result is an efficient method for computing the order of the group of rational points on the Jacobian of a smooth geometrically connected projective curve over a finite field of small characteristic.

Counting points on varieties over finite fields of small characteristic

We present a polynomial-time algorithm for computing the zeta function of a smooth projective hypersurface of degree d over a finite field of characteristic p, under the assumption that p is a suitably small odd prime and does not divide d. This improves significantly upon an earlier algorithm of the author and Wan which is only polynomial-time when the dimension is fixed.

/pdf/counting-solutions-to-equations-in-many-variables-over-3w1h1qeaiv.pdf

Alan G. B. Lauder

Papers

Deformation Theory and The Computation of Zeta Functions

Decomposition of Polytopes and Polynomials

Stark points and -adic iterated integrals attached to modular forms of weight one

Counting points on varieties over finite fields of small characteristic

Counting Solutions to Equations in Many Variables over Finite Fields