The Zariski topology on these objects is useful when studying the correspondence to ideals provided by Hilbert’s Nullstellensatz (“NOOL-shtell-in-zotts”). For 10 points each:
[10m] Name these central objects of the “geometry” side of algebraic geometry. Alexander Grothendieck introduced schemes as a generalization of these objects.
ANSWER: algebraic varieties [or algebraic variety; accept affine varieties or projective varieties]
[10e] Algebraic varieties are generated as the vanishing locus of these functions over an algebraically closed field. The Nullstellensatz generalizes the fundamental theorem of algebra, which concerns the roots of these functions.
ANSWER: polynomials
[10h] In a variant of algebraic geometry described by this adjective, varieties resemble linear meshes. That field studies semirings named for this adjective, which are obtained by treating classical addition as a product and introducing the minimum as a sum.
ANSWER: tropical [accept tropical geometry]
<JC, Other Science (Math)>