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1 (15/01)
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Introduction. Overview of the module, history of Gröbner bases. Examples of systems of polynomial equations in various research areas. Online interface to Magma, and some first computations.
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2 (16/01)
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First steps: monomials, polynomials. Two problems of basic commutative algebra: ideal membership and description of the quotient ring. Monomial orderings. Examples of monomial orderings: GLEX and LEX. Leading monomials, leading terms, leading coefficients. The space of leading terms.
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3 (19/01)
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No lecture on this day.
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4 (22/01)
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The space of leading terms is an ideal. Definition of a Gröbner basis. Examples. A Gröbner basis of an ideal generates that ideal. Gröbner bases and solutions to the two basic problems of commutative algebra.
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5 (23/01)
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Minimal Gröbner bases. Reduced Gröbner bases. Every ideal has a reduced Gröbner basis. Dickson's lemma. The reduced Gröbner basis of every ideal of the polynomial ring in n variables is finite. (Strong form of the Hilbert's Basis Theorem; Noetherianity of polynomial rings.)
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6 (26/01)
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S-polynomials. The Diamond Lemma for commutative polynomials (statement and proof).
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[HW1 (PDF)]
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7 (29/01)
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Buchberger's algorithm. Example. Termination of the algorithm (using the ACC version of Noetherianity). Remarks on the occasional double-exponential 'explosion' of Gröbner bases.
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8 (30/01)
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Optimisation of Buchberger's algorithm. Co-prime leading monomials create a redundant S-polynomial. The Triangle Lemma. Examples.
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9 (02/02)
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System of polynomial equations. Hilbert's Nullstellensatz (statement; proof over complex numbers, or any uncountable algebraically closed field). Gröbner bases and solutions of polynomial equations: Shape Lemma (statement).
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10 (05/02)
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Finiteness of the solution set for a system of polynomial equations and the dimension of the quotient ring by the ideal generated by equations. Shape Lemma (proof). First steps in elimination theory: elimination ideals of an ideal and their Gröbner bases for the lexicographic order. Example of the equations xy-1 and xz-1.
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[HW1 solutions (PDF)]
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11 (06/02)
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Elimination theory continued. Slogan: "Gröbner bases compute all elimination ideals in one go". Extension theorem. Statement and sketch of the proof. Elimination theory of the 19th century: the Sylvester matrix of two univariate polynomials, and its relationship to classifying common factors of those polynomials.
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[HW2 (PDF)]
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12 (09/02)
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Resultants and their properties. End of proof of Extension theorem: impossibility of the non-extendable case.
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13 (12/02)
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Two examples: equation for the parametric curve x=2t-4t3, y=t2-3t4 and the surjectivity of parametrisation in the complex and real case, Lagrange multipliers for the distance between an ellipse and a hyperbola.
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14 (13/02)
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Noncommutative polynomials. Some warnings on what noncommutativity brings (non-Noetherianity; even principal ideals are complicated). Monomial orderings. Definition of a Gröbner basis of a two-sided ideal. First properties of Gröbner bases. Small common multiples. Definition of an S-polynomial for a small common multiple.
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15 (16/02)
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Examples of S-polynomials. The Diamond Lemma for noncommutative polynomials.
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16 (19/02)
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Reduced Gröbner bases. Noncommutative Buchberger's "algorithm". Examples. Applications in group theory.
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17 (20/02)
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Statement and proof of the Poincaré-Birkhoff-Witt theorem.
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[HW3 (PDF)]
[HW2 solutions (PDF)]
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18 (23/02)
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Computing the number of normal monomials. Examples. Graph of normal monomials.
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Reading week, no classes
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19 (05/03)
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Graph of normal monomials. Paths in that graph and counting normal monomials. Counting number of paths in a directed graph, and application of Cayley-Hamilton theorem.
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20 (06/03)
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Linear recurrence for the numbers of normal monomials, and rationality of the corresponding power series. Example of an ideal that has an infinite Gröbner basis for every monomial ordering.
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21 (09/03)
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Example of an ideal that has infinitely many ideals of leading terms for various monomial orderings.
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22 (12/03)
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The set of all total orders on a set S as a topological space. For a countable set, that topology is induced by a metric.
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23 (13/03)
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The set of all total orders on a countable set S is a compact topological space. The set of monomial orderings is a closed subset of the set of total orders. Equivalent definition of a monomial ordering: the property 1≤m for all m implies the well-order property.
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[HW4 (PDF)]
[HW3 solutions (PDF)]
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24 (16/03)
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Every ideal in the commutative polynomial ring has a finite universal Gröbner basis. An uncountable family of monomial orderings. Classification of monomial orderings in the commutative case (statement).
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25 (19/03)
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Saint Patrick's day observed: public holiday, no lecture
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26 (20/03)
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No lecture on this day.
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27 (23/03)
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Ordered fields. Examples and non-examples. Non-archimedean ordering on the field of rational functions with coefficients in the ordered field. Classification of monomial orderings in the commutative case (beginning of proof).
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28 (26/03)
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Classification of monomial orderings in the commutative case (end of proof). Revision of the module.
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29 (27/03)
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Revision of the module. An example for the noncommutative Buchberger algorithm and the graph of normal monomials: the ideal (yz-x2, zx-y2, xy-z2) in F[x,y,z]. Noncommutative triangle lemma (reference: Prop. 2.4.3.2 in [BD]).
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[HW4 solutions (PDF)]
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30 (30/03)
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Good Friday: public holiday, no lecture
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