\protect \contentsline {section}{\protect \numberline {1}Announcements}{1}
\protect \contentsline {section}{\protect \numberline {2}Key ideas from last time}{2}
\protect \contentsline {section}{\protect \numberline {3}References for today's lecture}{2}
\protect \contentsline {section}{\protect \numberline {4}Gr\protect \relax \protect \"obner\ bases and systems of polynomial equations}{2}
\protect \contentsline {subsection}{\protect \numberline {4.1}Definitions}{3}
\protect \contentsline {section}{\protect \numberline {5}Recipes}{4}
\protect \contentsline {subsection}{\protect \numberline {5.1}Recipe 0---Main property}{4}
\protect \contentsline {subsection}{\protect \numberline {5.2}Recipe 1---existence of solutions}{5}
\protect \contentsline {subsection}{\protect \numberline {5.3}Recipe 2---existence of a finite number of solutions}{5}
\protect \contentsline {subsection}{\protect \numberline {5.4}Recipe 3---reduction to triangular form}{5}
\protect \contentsline {subsection}{\protect \numberline {5.5}Recipe 4---the Shape Lemma}{6}
\protect \contentsline {section}{\protect \numberline {6}Recipe 5---Linear Algebra Methods}{7}
\protect \contentsline {subsection}{\protect \numberline {6.1}Some more definitions}{7}
\protect \contentsline {subsection}{\protect \numberline {6.2}Multivariate division by an ideal}{7}
\protect \contentsline {subsection}{\protect \numberline {6.3}S-polynomials and Buchberger's algorithm}{8}
\protect \contentsline {subsubsection}{\protect \numberline {6.3.1}Buchberger's Algorithm}{9}
\protect \contentsline {subsection}{\protect \numberline {6.4}The vector space of reduced polynomials}{10}
\protect \contentsline {subsection}{\protect \numberline {6.5}Roots of the system are eigenvalues of a commuting family}{11}