\contentsline {section}{\numberline {1}Announcements}{1}
\contentsline {section}{\numberline {2}Key ideas from last time}{2}
\contentsline {section}{\numberline {3}References for today's lecture}{2}
\contentsline {section}{\numberline {4}Gr{\accent "7F o}bner\ bases and systems of polynomial equations}{2}
\contentsline {subsection}{\numberline {4.1}Definitions}{3}
\contentsline {section}{\numberline {5}Recipes}{4}
\contentsline {subsection}{\numberline {5.1}Recipe 0---Main property}{4}
\contentsline {subsection}{\numberline {5.2}Recipe 1---existence of solutions}{5}
\contentsline {subsection}{\numberline {5.3}Recipe 2---existence of a finite number of solutions}{5}
\contentsline {subsection}{\numberline {5.4}Recipe 3---reduction to triangular form}{5}
\contentsline {subsection}{\numberline {5.5}Recipe 4---the Shape Lemma}{6}
\contentsline {section}{\numberline {6}Recipe 5---Linear Algebra Methods}{7}
\contentsline {subsection}{\numberline {6.1}Some more definitions}{7}
\contentsline {subsection}{\numberline {6.2}Multivariate division by an ideal}{7}
\contentsline {subsection}{\numberline {6.3}The vector space of reduced polynomials}{8}
\contentsline {subsection}{\numberline {6.4}Roots of the system are eigenvalues of a commuting family}{9}
\contentsline {section}{\numberline {7}A real definition of a Gr{\accent "7F o}bner\ basis}{10}