# Computer Algebra Lecture Log (2018-2019)

This contains a brief report of each of the lectures that have taken place. It will be updated soon after each lecture.

Note: Last year's entries are left here so that you can see what will be covered and when (though there will be minor differences). As each entry is updated the correct date will be shown against it and the horizontal rule will be below the latest updated entry.

• Lecture 1 (14/01/2019).
• Handouts: Lecture notes pp. 1-42. Exercises 1.
• Topics covered: Introduction to the course. Reasons for creating computer algebra systems and need for sound foundations. Illustrative examples and features of systems. Brief forward look at Exercises 2 (finding the common points of two algebraic curves) as a motivating example.
• Notes: pp. 1-11 for home study (some of the material on Axiom is helpful for Exercises 1).
• Lecture 2 (17/01/2019).
• Handouts: None.
• Topics covered: Rings, conventions and simple properties. Commutative rings, multiplicative identities, inverses. Various short proofs given, e.g., uniqueness of 0 in a ring, uniqueness of additive inverses, uniqueness of multiplicative identity and inverses (when they exist, exercise). Looked at the integers modulo 6 and the fact that two non-zero elements can give zero when multiplied. Fields, proof that in a field multiplying non-zero elements yields a non-zero result.
• Suggested exercise: 4.2.
• Notes: pp. 12-15.
• Lecture 3 (21/01/2019).
• Handouts: None.
• Topics covered: Discussion of suggested exercise from previous lecture. Integral domains and (briefly) unique factorisation domains. Greatest common divisors. Canonical and normal representations. Integers. Karatsuba's algorithm (quick description, details in the notes).
• Notes: pp. 16-19.
• Lecture 4 (24/01/2019).
• Handouts: None.
• Topics covered: Rationals. Sensible approach to arithmetic on rationals. Euclid's algorithm for the integers. Extended version of Euclid's Algorithm, application: proof that Z_n is a field if and only if n is a prime. Polynomial rings in one indeterminate and in several indeterminates. Conventions and definitions: degree etc. Difference between polynomials and polynomial functions; significance of this for algorithmic work.
• Suggested exercises: 4.5, 5.8.
• Notes: pp. 19-23, 26-29, 31-32 (omit sections 4.6.2, 4.7.2,).
• Lecture 5 (28/01/2019).
• Handouts: None.
• Topics covered: Discussion of suggested exercises. Factorisation and gcd's, fact that a polynomial ring over a UFD is itself a UFD. Defining the gcd of polynomials by means of common divisors of highest degree. Relevance of degree for gcds of univariate polynomials with coefficients from a field as opposed to a UFD such as the integers. Euclid's algorithm for univariate polynomials, Division with remainder of univariate polynomials (exercise). Uniqueness of quotient and remainder in polynomial division (exercise). Discussion of efficiency issues for Euclid's algorithm when rational coefficients are involved.
• Suggested exercise: Prove that quotient and remainder are unique for polynomial division.
• Notes: pp. 33-37. Omit section 4.7.4.
• Lecture 6 (31/01/2019).
• Handouts: None.
• Topics covered: Discussion of suggested exercise. Extended Euclidean algorithm for polynomials. Rational expressions, discussion of subtleties relating to defining binary operations. Representations of polynomials. Representation of rational expressions using sums and products. Intermediate expressions swell and implications for CA systems when dividing (detecting division by 0). Computing gcds for univariate polynomials with integer coeficients. Gauss' Lemma and its application to the problem.
• Suggested exercise: Prove that the a monic gcd of univariate polynomials with coefficients form a field is unique.
• Notes: pp. 37-43. Omit Section 4.7.7
• Lecture 7 (04/02/2019).
• Handouts: Exercises 2. Notes pp. 43-57.
• Topics covered: Discussion of suggested exercise. Notion of doing things modulo a prime. Discussion on how to generalise the modular method, problems to address. Started detailed example of a gcd calculation for two primitive polynomials with integer coefficients.
• Suggested exercises: Let f, g be polynomials in x with integer coefficients. Let h be their gcd in Z[x]. Prove that h is also a gcd of f, g in Q[x].
• Notes: pp. 43-45.
• Lecture 8 (07/02/2019).
• Handouts: None.
• Topics covered: Discussion of suggested exercises.
• Notes: Completed detailed example of a gcd calculation for two primitive polynomials with integer coefficients. Discussed various issues raised and their significance (e.g., that it would be nonsense to try to compute a gcd in Z_35[x]). The Chinese remainder Theorem and its use for recovering integer coefficients in polynomial gcd calculations. Symmetric representation of remainders. Bounds on the coefficients of gcd's (outline of result only, no proof).
• Suggested exercises: Prove that if m1, m2 and m3 are coprime then m1m2 and m3 are coprime. Prove the claims of Theorem 5.3.
• Notes: pp. 45-49.
• Lecture 9 (11/02/2019).
• Handouts: None.
• Topics covered: Discussion of suggested exercises. Choosing good primes; the resultant. Properties of the resultant; finiteness of number of bad primes. Full modular gcd algorithm with comments on implementation heuristics. Final example.
• Notes: pp. 50-55. Section 5.3 need not be studied only the fact that bounds can be found needs to be known.
• Lecture 10 (14/02/2019).
• Handouts: Lecture notes pp. 58-89.
• Topics covered: Introduction to varieties and ideals. Examples of ideals and connection with equations. Hilbert's Basis Theorem and its significance. Discussion of ideals and their properties. Going from ideals to varieties and vice versa. Algebraically closed fields. Hilbert's Nullstellensatz.
• Suggested exercises: Prove various simple properties of ideals. Prove that I(V) is an ideal for any set V. Prove that every algebraically closed field is infinite.
• Notes: pp. 58-63. Proof of HBT is optional.
• Lecture 11 (25/02/2019).
• Handouts: None.
• Topics covered: Discussion of various forms of Hilbert's Nullstellensatz. Polynomial Ideal Membership problem. Discussion of how one might solve the Polynomial Ideal Membership problem: notion of reduction sequences and the restricted version (also viewed as a form of division). Discussion of restricted reduction algorithm and example to show that it is not guaranteed to work. Discussion of what goes wrong with restricted reductions approach and how to mend it leading to the definition of Groebner bases for a non-zero ideal.
• Suggested exercise: Let F be a set of polynomials and f, g be two polynomials such that f reduces to g w.r.t. F. Prove that f is in the ideal (F) if and only if g is in the ideal (F).
• Notes: pp. 63-66.
• Lecture 12 (28/02/2019).
• Handouts: None.
• Topics covered: Discussion of suggested exercises. Sketch proof of the existence of Groebner bases. Discussion of how to design an algorithm to compute Groebner bases, notion of S-polynomial. Formal definition and characterisation of Groebner bases, admissible orders. Proof that an admissible order cannot have infinitely decreasing sequences.
• Suggested exercise: Let S be a set of power products in k[X] and I=(S). Prove that (i) a power product u is in I if and only if it is divisible by some s from S. (ii) a polynomial f=c_1s_1+..+c_rs_r where each c_i is a non-zero constant and each s_i is a power product, belongs to I if and only if each power product belongs to I.
• Notes: pp. 66-70.
• Lecture 13 (04/03/2019).
• Handouts: Exercises 3.
• Topics covered: Discussion of suggested exercise. Proof that reductions always terminate. Proof that a set of monomialsis automatically a Groebner basis for the ideal it generates. Algorithm for Groebner bases, example. Proof that Groebner basis algorithm terminates. Notions of minimal, reduced and normed bases, proof of their uniqueness.
• Notes: pp. 70-76. (Omit Lemmas 6.4, 6.5, Theorem 6.4.) (Lemma 6.3 and Theorem 6.3 state things that were covered during the development part).
• Lecture 14 (07/03/2019).
• Handouts: None.
• Topics covered: Applications of Groebner bases, ideal membership, solution of equations. Deciding if a system has finitely many solutions; diagonalisation of systems of equations. Proof of diagonalisation theorem for Groebner bases. Brief (non-examinable) discussion of worst case cost of computing Groebner bases (no proofs), setting it in context for applications.
• Notes: pp. 76-78,
• Lecture 15 (11/03/2019).
• Handouts: Lecture notes pp. 90-121.
• Topics covered: Real roots of univariate polynomials. Notions of isolation and approximation. Importance of square free polynomials in detecting roots. Computing square free parts of polynomials. Simple approximation algorithm (given an isolating interval). Isolation algorithm, given a means of counting roots. Bounds on the roots of a polynomial.
• Notes: pp. 90-95.
• Lecture 16 (14/03/2019).
• Handouts: None.
• Topics covered: Variations of sequences leading to Sturm's method for counting roots. Sturm sequences and simple properties.Proof of correcness of Sturm's method. Simple examples. Examples. Method of counting all the real roots without using bounding intervals. Brief discussion of efficiency problems with Sturm sequences.
• Notes: pp. 98-100.
• Lecture 17 (18/03/2019).
• Handouts: Guide to revision (on web page).
• Topics covered: Method by continued fractions. Outline of method using an example. Completed outline of real root isolation by continued fractions (non-examinable). Moebius transforms. Theorem of Vincent. Proof that every polynomial whose coefficients have exactly one sign variation has exactly one strictly positive real root. Experimental results and some result results on the breadth of the associated tree.
• Notes: pp. 102-108 (all non-examinable)
• Lecture 18 (22/03/2019).
• Handouts: Guide to revision.
• Topics covered: Discussed the course aims and objectives, approach to revision and exam. Question and answer session.
• Notes: None.

Kyriakos Kalorkoti, IF5.26a

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