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

Introduction to finite fields and their applications: Preface

INTRODUCTION TO QUADRATIC FORMS OVER FIELDS (Graduate Studies in Mathematics 67) By T. Y. LAM: 550 pp., US$79.00, ISBN 0-8218-1095-2 (American Mathematical Society, Providence, RI, 2005)

Cameron-Liebler line classes with parameter x = q 2 - 1 2

In these lecture notes we give sketches of classical undecidability results in number theory, like Godel’s first Incompleteness Theorem (that the first order theory of the integers in the language of rings is undecidable), Julia Robinson’s extensions of this result to arbitrary number fields and rings of integers in them, as well as to the ring of totally real integers, and Matiyasevich’s negative solution of Hilbert’s 10th problem, i.e., the undecidability of the existential first-order theory of the integers. As Hilbert’s 10th problem is still open for the rationals (i.e., the question whether the existential theory of the field of rational numbers is decidable) we also present a sketch of the fact that there is a universal definition of the ring of integers inside the field of rationals. In terms of complexity this is the simplest definition known so far. If one had an existential definition instead then Hilbert’s 10th problem over the rationals would reduce to that over the integers (and hence be, as expected, unsolvable), but, modulo a widely believed in conjecture in number theory, we also indicate why there should be no such existential definition. We conclude with a list of nice open questions in the area.

Undecidability in Number Theory

In this paper, we describe a new infinite family of $$\frac{q^{2}-1}{2}$$q2-12-tight sets in the hyperbolic quadrics $${\mathcal {Q}}^{+}(5,q)$$Q+(5,q), for $$q \equiv 5 \text{ or } 9 \,\hbox {mod}\,{12}$$q?5or9mod12. Under the Klein correspondence, these correspond to Cameron---Liebler line classes of $$\mathop {\mathrm{PG}}(3,q)$$PG(3,q) having parameter $$\frac{q^{2}-1}{2}$$q2-12. This is the second known infinite family of nontrivial Cameron---Liebler line classes, the first family having been described by Bruen and Drudge with parameter $$\frac{q^{2}+1}{2}$$q2+12 in $$\mathop {\mathrm{PG}}(3,q)$$PG(3,q) for all odd $$q$$q. The study of Cameron---Liebler line classes is closely related to the study of symmetric tactical decompositions of $$\mathop {\mathrm{PG}}(3,q)$$PG(3,q) (those having the same number of point classes as line classes). We show that our new examples occur as line classes in such a tactical decomposition when $$q \equiv 9 \,\hbox {mod}\,12$$q?9mod12 (so $$q = 3^{2e}$$q=32e for some positive integer $$e$$e), providing an infinite family of counterexamples to a conjecture made by Cameron and Liebler (in Linear Algebra Appl 46, 91---102, 1982); the nature of these decompositions allows us to also prove the existence of a set of type $$\left( \frac{1}{2}(3^{2e}-3^{e}), \frac{1}{2}(3^{2e}+3^{e}) \right) $$12(32e-3e),12(32e+3e) in the affine plane $$\mathop {\mathrm{AG}}(2,3^{2e})$$AG(2,32e) for all positive integers $$e$$e. This proves a conjecture made by Rodgers in his Ph.D. thesis.

A new family of tight sets in $${\mathcal {Q}}^{+}(5,q)$$Q+(5,q)

In this paper, we describe a new infinite family of $\frac{q^{2}-1}{2}$-tight sets in the hyperbolic quadrics $\mathcal{Q}^{+}(5,q)$, for $q \equiv 5 \mbox{ or } 9 \bmod{12}$. Under the Klein correspondence, these correspond to Cameron--Liebler line classes of ${\rm PG}(3,q)$ having parameter $\frac{q^{2}-1}{2}$. This is the second known infinite family of nontrivial Cameron--Liebler line classes, the first family having been described by Bruen and Drudge with parameter $\frac{q^{2}+1}{2}$ in ${\rm PG}(3,q)$ for all odd $q$. 
The study of Cameron--Liebler line classes is closely related to the study of symmetric tactical decompositions of ${\rm PG}(3,q)$ (those having the same number of point classes as line classes). We show that our new examples occur as line classes in such a tactical decomposition when $q \equiv 9 \bmod 12$ (so $q = 3^{2e}$ for some positive integer $e$), providing an infinite family of counterexamples to a conjecture made by Cameron and Liebler in 1982; the nature of these decompositions allows us to also prove the existence of a set of type $\left(\frac{1}{2}(3^{2e}-3^{e}), \frac{1}{2}(3^{2e}+3^{e}) \right)$ in the affine plane ${\rm AG}(2,3^{2e})$ for all positive integers $e$. This proves a conjecture made by Rodgers in his PhD thesis.

A new family of tight sets in $\mathcal{Q}^{+}(5,q)$

We present various facts on the graded Betti table of a projectively embedded toric surface, expressed in terms of the combinatorics of its defining lattice polygon. These facts include explicit formulas for a number of entries, as well as a lower bound on the length of the quadratic strand that we conjecture to be sharp (and prove to be so in several special cases). We also present an algorithm for determining the graded Betti table of a given toric surface by explicitly computing its Koszul cohomology and report on an implementation in SageMath. It works well for ambient projective spaces of dimension up to roughly 25, depending on the concrete combinatorics, although the current implementation runs in finite characteristic only. As a main application we obtain the graded Betti table of the Veronese surface nu(6)(P-2) subset of P-27 in characteristic 40 009. This allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary Veronese surface nu(d)(P-2).

Computing graded Betti tables of toric surfaces

We construct a Diophantine interpretation of $\mathbb{F}_q[W,Z]$
 over $\mathbb{F}_q[Z]$
 . Using this together with a previous result that every recursively enumerable (r.e.) relation over $\mathbb{F}_q[Z]$
 is Diophantine over $\mathbb{F}_q[W,Z]$
 , we will prove that every r.e. relation over $\mathbb{F}_q[Z]$
 is Diophantine over $\mathbb{F}_q[Z]$
 . We will also look at recursive infinite base fields $\mathbb{F}$
 , algebraic over $\mathbb{F}_p$
 . It turns out that the Diophantine relations over $\mathbb{F}[Z]$
 are exactly the relations which are r.e. for every recursive presentation.

/pdf/recursively-enumerable-sets-of-polynomials-over-a-finite-2fb171k7lp.pdf

Recursively enumerable sets of polynomials over a finite field are diophantine

https://biblio.ugent.be/publication/373953/file/734806.pdf

Jeroen Demeyer

Papers

A new family of tight sets in $${\mathcal {Q}}^{+}(5,q)$$Q+(5,q)

A new family of tight sets in $\mathcal{Q}^{+}(5,q)$

Computing graded Betti tables of toric surfaces

Recursively enumerable sets of polynomials over a finite field are diophantine

Recursively enumerable sets of polynomials over a finite field