Hilbert functions and resolutions are both central objects in
commutative algebra and fruitful tools in the fields of algebraic
geometry, combinatorics, commutative algebra, and computational
algebra. Spurred by recent research in this area, Syzygies and
Hilbert Functions explores fresh developments in the field as well
as fundamental concepts. Written by international mathematics
authorities, the book first examines the invariant of
Castelnuovo-Mumford regularity, blowup algebras, and bigraded
rings. It then outlines the current status of two challenging
conjectures: the lex-plus-power (LPP) conjecture and the
multiplicity conjecture. After reviewing results of the geometry of
Hilbert functions, the book considers minimal free resolutions of
integral subschemes and of equidimensional Cohen-Macaulay
subschemes of small degree. It also discusses relations to subspace
arrangements and the properties of the infinite graded minimal free
resolution of the ground field over a projective toric ring. The
volume closes with an introduction to multigraded Hilbert
functions, mixed multiplicities, and joint reductions. By surveying
exciting topics of vibrant current research, Syzygies and Hilbert
Functions stimulates further study in this hot area of mathematical
activity.
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