Basic Algebraic Geometry

By Andrew J. Sommese, Charles W. Wampler II. 2005

762 citations recorded by the source corpus.

Subfields: Biomedical Engineering, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0a539ff462d0d6ca7cb0e7e6c7c26f6bd768f430

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In Biomedical Engineering

In Geometry and Topology

Citations in the corpus

Piecewise linear differential systems with only centers can create limit cycles? (2017, background): “We recall that Bezout Theorem (see for instance [26]) states that if a polynomial differential system of equations has finitelymany solutions, then the number of its solutions is at most the product of the degrees of the polynomials which appear in the system.”
Symmetric Tensors and Symmetric Tensor Rank (2008, neutral): “This justifies the use of the Zariski topology, where the elementary closed subsets are the common zeros of a finite number of homogeneous polynomials [48].”
Sums of Squares, Moment Matrices and Optimization Over Polynomials (2009, neutral): “[150]).”
A multi-dimensional resolution of singularities with applications to analysis (2010, methodology): “A transformation of this form will be referred to as a blow-down, in keeping with the standard nomenclature of these maps in the algebraic geometry and microlocal analysis literature [50].”
Krull dimension of solvable groups (2008, background): “In algebraic geometry the typical topological spaces are Noetherian, in which case the notion of Zariski dimension arises naturally [28, 56, 19] and gives the dimension of irreducible varieties in affine spaces.”
Galois Theory of Linear Differential Equations (2012, background): “Although we present the following material ab initio, the reader completely unfamiliar with most of the above ideas of algebraic geometry would profit from looking at [75] or the introductory chapters of [124], [213] or [261].”
Topology of character varieties of Abelian groups (2013, background): “surjective ψ(Cr,n) = Cr. Nowsuppose thatCr,n isirreducible. Then, clearly Hom(Zr,GL(n,C)) is dense in Cr,n, being the complement of the trivial determinant subvariety. Moreover, by Bertini’s theorem ([44], vol I, p. 139) there is a Zariski open subset U ⊂ Cr such that ψ−1(y) is irreducible for y ∈ U”
NONCOMMUTATIVE CONVEXITY ARISES FROM LINEAR MATRIX INEQUALITIES. (2006, background): “Also beware that a standing hypothesis in this part of [32] is that the base field is algebraically closed.”