Seminar in Algebra

Cox, David, John Little, and Donal O’Shea. Ideals, Varieties, and Algorithms . 3rd ed. Undergraduate Texts in Mathematics. New York, NY: Springer, 2007. ISBN: 9780387356518.

The topics follow the order of the required textbook:

SES #: 1 · TOPICS: Polynomials and affine space, affine varieties · READINGS: Sections 1-1 and 1-2

SES #: 2 · TOPICS: Parameterizations of affine varieties, ideals · READINGS: Sections 1-3 and 1-4

SES #: 3 · TOPICS: Polynomials of one variable, orderings on the monomials in k [ x 1 ,…, x n ] · READINGS: Sections 1-5, 2-1, and 2-2

SES #: 4 · TOPICS: A division algorithm in k [ x 1 ,…, x n ], monomial ideals and Dickson’s lemma · READINGS: Sections 2-3 and 2-4

SES #: 5 · TOPICS: The Hilbert basis theorem and Groebner bases, properties of Groebner bases · READINGS: Sections 2-5 and 2-6

SES #: 6 · TOPICS: Buchberger’s algorithm, first applications of Groebner bases · READINGS: Sections 2-7 and 2-8

SES #: 7 · TOPICS: The elimination and extension theorems, the geometry of elimination · READINGS: Sections 3-1 and 3-2

SES #: 8 · TOPICS: Implicitization, singular points and envelopes · READINGS: Sections 3-3 and 3-4

SES #: 9 · TOPICS: Unique factorization and resultants · READINGS: Section 3-5

SES #: 10 · TOPICS: Resultants and the extension theorem, the nullstellensatz · READINGS: Sections 3-6 and 4-1

SES #: 11 · TOPICS: Radical ideals and the ideal-variety correspondence, sums, products, and intersections of ideal · READINGS: Sections 4-2 and 4-3

SES #: 12 · TOPICS: Zariski closure and quotients of ideals, irreducible varieties and prime ideals · READINGS: Sections 4-4 and 4-5

SES #: 13 · TOPICS: Decomposition of a variety into irreducibles, polynomial mappings · READINGS: Sections 4-6 and 5-1

SES #: 14 · TOPICS: Quotients of polynomials R, algorithmic computations in k [ x 1 ,…, x n ]/ I · READINGS: Sections 5-2 and 5-3

SES #: 15 · TOPICS: The coordinate ring of an affine variety, rational functions on a variety · READINGS: Sections 5-4 and 5-5

SES #: 16 · TOPICS: Proof of the Closure theory, geometric description of robots, the forward kinematics problem · READINGS: Sections 5-6, 6-1, and 6-2

SES #: 17 · TOPICS: The inverse kinematic problem and motion planning, automatic geometric theorem proving · READINGS: Sections 6-3 and 6-4

SES #: 18 · TOPICS: Wu’s method, symmetric polynomials · READINGS: Sections 6-5 and 7-1

SES #: 19 · TOPICS: Finite matrix groups and rings of invariants, generators for the ring of invariants · READINGS: Sections 7-2 and 7-3

SES #: 20 · TOPICS: Relations among generators and the geometry of orbits, the projective plane, projective space and projective varieties · READINGS: Sections 7-4, 8-1, and 8-2

SES #: 21 · TOPICS: The projective algebra-geometry dictionary, the projective closure of an affine variety · READINGS: Sections 8-3 and 8-4

SES #: 22 · TOPICS: Projective elimination theory · READINGS: Section 8-5

SES #: 23 · TOPICS: The geometry of quadric hypersurfaces, the variety of a monomial ideal · READINGS: Sections 8-6 and 9-1

SES #: 24 · TOPICS: The complement of a monomial ideal, the Hilbert function and the dimension of a variety · READINGS: Sections 9-2 and 9-3

SES #: 25 · TOPICS: Elementary properties of dimension, dimension and algebraic independence · READINGS: Sections 9-4 and 9-5

SES #: 26 · TOPICS: Dimension and nonsingularity, the tangent cone · READINGS: Sections 9-6 and 9-7

Silverman, Joseph H., and John Tate. Rational Points on Elliptic Curves . New York: Springer-Verlag, 1 August 1992. ISBN: 0387978259.

The last two lectures are based on the textbook below:

Koblitz, Neal. Introduction to Elliptic Curves and Modular Forms . 2nd ed. New York: Springer-Verlag, 1993. ISBN: 0387979662.

Most of the material for the class will follow the text: Silverman, Joseph H., and John Tate. Rational Points on Elliptic Curves. New York: Springer-Verlag, 1 August 1992. ISBN: 0387978259.

The last two lectures are based on the textbook: Koblitz, Neal. Introduction to Elliptic Curves and Modular Forms. 2nd ed. New York: Springer-Verlag, 1993, ISBN: 0387979662.

lec #: 1 · Topics: The Projective Plane · Readings: Section A.1: The Projective Plane · NOTES: First Half: Section A.1, up through p. 222 line 11 “… coordinates for the point.”

lec #: 2 · Topics: Curves in the Projective Plane · Readings: Section A.2: Curves in the Projective Plane · NOTES: First Half: Section A.2, up through p. 228 line 4 (the displayed equations.)

lec #: 3 · Topics: Rational Points on Conics · Readings: Section I.1: Rational Points on Conics · NOTES: First Part: Section I.1. up through line 4 from the bottom of p. 11 “…infinity for t!]”

lec #: 4 · Topics: Geometry of Cubic Curves · Readings: Section I.2: Geometry of Cubic Curves · NOTES: First Part: Section I.2, up through p. 18 second full paragraph, ending “…is shown in Figure 7.” If this is too much for 25 minutes, consider stating but omitting the proof of the theorem concerning 3 cubic curves which is stated at the bottom of p. 16. In other words, you could say we accept this theorem without proof and omit the argument in the first 3 paragraphs on p. 17.

lec #: 5 · Topics: Weierstrass Normal Form · Readings: Section I.3: Weierstrass Normal Form · NOTES: First Part: Section I.3, up through p. 24 line 3 from the end “…points on cubic curves in Weierstrass normal form.” You will probably want to omit some details here and there to make it fit to 25 minutes.

lec #: 6 · Topics: Explicit Formulas for the Group Law · Readings: Section I.4: Explicit Formulas for the Group Law · NOTES: Please cover as much of Section I.4 as seems reasonable in 30 minutes or so. You can concentrate on the formula for adding distinct points, and not worry about the doubling formula, unless you have time. A specific example is also called for.

lec #: 7 · Topics: Points of Order Two and Three · Readings: Section II.1: Points of Order Two and Three · NOTES: You should be able to cover the entire contents of Section II.1 comfortably in 50 minutes. Take your time.

lec #: 9 · Topics: Points of Finite Order have Integer Coordinates - Part 2 · Readings: Section II.4: Points of Finite Order have Integer Coordinates - Part 2 · NOTES: This lecture contains a lot of technical material in the middle of Section II.4 (sorry.) Start by reviewing the definition of C(p^{nu}) in the second full paragraph on p. 50 (or make this definition for the first time if the previous speaker didn’t get to it.) Then continue through to p. 54 up through paragraph 4 ending “….t(P)= x(P)/y(P).”

lec #: 11 · Topics: Real and Complex Points on Cubics · Readings: Section II.2: Real and Complex Points on Cubics · NOTES: This section is kind of difficult, and we will not really use it much later. But its ideas are very important so I want everyone to be exposed to them, even if not everyone understands everything here.

lec #: 12 · Topics: Heights and Descent · Readings: Section III.1: Heights and Descent · NOTES: This section, Section III.1, is possibly a bit long for one lecture. Please state the descent theorem fully on the board. You can definitely cut out the last paragraph on p. 67 and p. 68 if necessary. Let me know if it still runs long.

lec #: 13 · Topics: Height of P + P_0 · Readings: Section III.2: Height of P + P_0 · NOTES: Please cover Section III.2 entirely. The argument looks harder than it is, because it has so much notation.

lec #: 14 · Topics: Height of 2P · Readings: Section III.3: Height of 2P · NOTES: Please cover section III.3. This is a lot of material for one lecture, but I don’t think the techniques are interesting enough to deserve being spread over 2 lectures. So here’s some advice.

lec #: 15 · Topics: A Useful Homomorphism - Part 1 · Readings: Section III.4: A Useful Homomorphism - Part 1 · NOTES: Starting with this section we will slow down a bit and take two lectures per section, both because the sections are longer and the techniques they introduce are more interesting.

lec #: 16 · Topics: A Useful Homomorphism - Part 2 · Readings: Section III.4: A Useful Homomorphism - Part 2 · NOTES: Please state the proposition on p. 79 and complete the proof of it, taking you up through the end of section III.4. The authors leave some bits of the proof to the reader. You can present some of those if you want.

lec #: 17 · Topics: Mordell’s Theorem - Part 1 · Readings: Section III.5: Mordell’s Theorem - Part 1 · NOTES: Please cover section III.5 on Mordell’s Theorem, through the statement of the Proposition on p. 85 and part of the proof. I don’t want to rush this, so just prove parts (a) and (b) of the proposition and leave the rest for the next person. This may give you time to be more detailed in your proof of parts (a) and (b) than the book.

lec #: 19 · Topics: Examples - Part 2 · Readings: Section III.6: Examples - Part 2 · NOTES: Continue Section III.6, beginning in the middle of p. 90 with the paragraph “In our case what are the possibilities…” Remind everyone a little of what happened last time, including what the notation #Gamma[2] means, and the formula for it which was the end of the last lecture.

lec #: 20 · Topics: Examples - Part 3 · Readings: Section III.6: Examples - Part 3 · NOTES: Present the examples 1-4 on pages 94-98. I will likely assign some homework exercises which require you all to do calculations like these to find the rank of certain elliptic curves. So take your time to try to make it clear what is going on in these examples. If that means only doing some of the examples (for example omitting example 4), that’s OK.

lec #: 21 · Topics: Singular Cubics · Readings: Section III.7: Singular Cubics · NOTES: Cover all of section III.7 on Singular Cubics. This is essentially a detour, since the nonsingular curves are the most interesting ones for us, but for completeness it is worthwhile to see what happens in the singular case. You should be able to cover this material in one class (omitting, just as the book, the proof of part (b) of the main theorem.)

lec #: 22 · Topics: Rational Points over Finite Fields · Readings: Section IV.1: Rational Points over Finite Fields · NOTES: Cover Section IV.1 in the book. In this chapter we will be working over the “finite field of p elements F_p” for some prime p. It is worth noting that all this means is that we are doing “arithmetic modulo p” exactly as in a first number theory course.

lec #: 23 · Topics: Gauss’s Theorem - Part 1 · Readings: Section IV.2: Gauss’s Theorem - Part 1 · NOTES: Cover Section IV.2 from the beginning up through the formula M_p = 9[RTS]/m on the top of p. 114. This is kinda dense, so time yourself beforehand to make sure it fits in 50 minutes. If not, you can assume some facts without proof, for example you could omit the proof that F_p^* is a cyclic group.

lec #: 24 · Topics: Gauss’s Theorem - Part 2 · Readings: Section IV.2: Gauss’s Theorem - Part 2 · NOTES: Continue in section IV.2 from the first full paragraph on p. 114 “Now we just have to find…” to the end of that section.

lec #: 25 · Topics: Points of Finite Order Revisited · Readings: Section IV.3: Points of Finite Order Revisited · NOTES: Do Section IV.3. This is an important section, since it gives us (yet another) improved method for finding rational points of finite order, which is useful for calculations.

lec #: 26 · Topics: Factorization using Elliptic Curves - Part 1 · Readings: Section IV.4: Factorization using Elliptic Curves - Part 1 · NOTES: Cover Section IV.4, up through the displayed line “p-1 = product of primes to small powers” on p. 132.

lec #: 27 · Topics: Factorization using Elliptic Curves - Part 2 · Readings: Section IV.4: Factorization using Elliptic Curves - Part 2 · NOTES: Describe the elliptic curve factoring algorithm in section IV.4. Specifically, begin on p. 132 with the paragraph “Now we are ready to describe Lenstra’s idea…” and go to the end of the section.

lec #: 30 · Topics: Thue’s Theorem - Part 2 · Readings: Section V.3: Thue’s Theorem - Part 2 · NOTES: State the Diophantine Approximation Theorem in Section V.3 on p. 153 (restate it if it is stated in the previous lecture), and continue through to the end of the section, which gives a sketch of the proof.

lec #: 31 · Topics: Construction of an Auxiliary Polynomial · Readings: Section V.4: Construction of an Auxiliary Polynomial · NOTES: You will handle the first step of the proof of the DAT, which is the construction of an Auxiliary polynomial in Section V.4. This section is problematic. It is very long, and there is a lot of notation, but overall it’s actually not doing anything all that complicated. Also, we don’t want to spread it over two lectures, because then the second lecture will be after Thanksgiving break.

lec #: 32 · Topics: The Auxiliary Polynomial is Small · Readings: Section V.5: The Auxiliary Polynomial is Small · NOTES: Do Section V.5 on the smallness of the Auxiliary polynomial. This is not really a long section, but it has a lot of notation. I think it’s worth doing in painful detail. Explain each step carefully, since you have time to do so. It’s probably worth keeping the Smallness Theorem up on the board, so we don’t forget where we’re going.

lec #: 33 · Topics: The Auxiliary Polynomial Does Not Vanish · Readings: Section V.6: The Auxiliary Polynomial Does Not Vanish · NOTES: Do Section V.6 on the non-vanishing of the auxiliary polynomial. This is not a huge amount of material for 50 minutes. So you should have time to begin with a reminder of how the Non-Vanishing Theorem generally fits in to our plan of proof of the DAT.

lec #: 35 · Topics: Congruent Numbers and Elliptic Curves I: Koblitz - Part 1 · Readings: Congruent Numbers and Elliptic Curves I: Koblitz, Section I.Intro, I.1. and I.2 · NOTES: In the last two lectures we will show how the material on elliptic curves we have already learned is related to an interesting problem in number theory, the congruent number problem.

Specifically, derive your lecture from the Chapter I introduction, section I.1, and section I.2 from Koblitz’s book. Most of the introduction can be skipped, though if you think you will have time please mention Tunnell’s theorem. Unfortunately, we won’t actually talk about that theorem again, so it is going to remain mysterious, I’m afraid.

lec #: 36 · Topics: Congruent Numbers and Elliptic Curves II: Koblitz - Part 2 · Readings: Congruent Numbers and Elliptic Curves II: Koblitz, Section I.9 · NOTES: In your lecture, you will complete the picture of exactly how the congruent number problem is related to the theory of elliptic curves, basing your lecture primarily on Section I.9 of Koblitz’s book.