Plane Algebraic Curves -- Ane Algebraic Curves -- Projective Algebraic Curves -- The Coordinate Ring of an Algebraic Curve and the Intersections of Two Curves -- Rational Functions on Algebraic Curves -- Intersection Multiplicity and Intersection Cycle of Two Curves -- Regular and Singular Points of Algebraic Curves. Tangents -- More on Intersection Theory. Applications -- Rational Maps. Parame…
Numerical Invariants of a Rees Algebra -- Hilbert Functions and Multiplicities -- Depth and Cohomology of Rees Algebras -- Divisors of a Rees Algebra -- Koszul Homology -- Integral Closure of Algebras -- Integral Closure and Normalization of Ideals -- Integral Closure of Modules -- HowTo.Integral Closure gives an account of theoretical and algorithmic developments on the integral closure of alg…
Field extensions -- Roots -- Galois theory -- A bit of group theory -- Applications -- Algebraic theory of differential equations.This unique textbook focuses on the structure of fields and is intended for a second course in abstract algebra. Besides providing proofs of the transcendance of pi and e, the book includes material on differential Galois groups and a proof of Hilbert's irreducibilit…
Monomial Ideals -- Squarefree monomial ideals -- Borel-fixed monomial ideals -- Three-dimensional staircases -- Cellular resolutions -- Alexander duality -- Generic monomial ideals -- Toric Algebra -- Semigroup rings -- Multigraded polynomial rings -- Syzygies of lattice ideals -- Toric varieties -- Irreducible and injective resolutions -- Ehrhart polynomials -- Local cohomology -- Determinants…