Open AccessBook
Complexity and Real Computation
TLDR
This chapter discusses decision problems and Complexity over a Ring and the Fundamental Theorem of Algebra: Complexity Aspects.Abstract:
1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem "P ? NP?".- 8 Newton's Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bezout's Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.read more
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Average volume, curvatures, and Euler characteristic of random real algebraic varieties
TL;DR: The expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution is determined, whose distribution is invariant under the action of the orthogonal group.
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Dimensional Differences between Nonnegative Polynomials and Sums of Squares
TL;DR: In this paper, the dimensions of the faces of nonnegative polynomials and the cone of sums of squares were studied, and it was shown that there are dimensional differences between corresponding faces of these cones.
Journal ArticleDOI
On the geometry of random lemniscates
TL;DR: The geometry of a random rational lemniscate is studied in this paper, where it is shown that the average spherical length of the lemnical function is proportional to the square root of the maximal spherical length.
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Condition length and complexity for the solution of polynomial systems
TL;DR: Beltran-Pardo and Burgisser-Cucker as mentioned in this paper showed that the complexity of finding a single zero of polynomial systems can be improved by using homotopy methods.
Journal ArticleDOI
Approximate Solutions for Abstract Inequality Systems
Chong Li,Kung Fu Ng +1 more
TL;DR: It is proved that, under the suitable conditions, the system of conic inequality systems of the type F(x) 0 is solvable, and the ratio of the distance from $x_0$ to the solution set $S$ over thedistance from F( x_0) to the cone $K$ has an upper bound given explicitly in terms of $\tau$ and $x-0".
References
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Book
Principles of Algebraic Geometry
Phillip Griffiths,Joe Harris +1 more
TL;DR: In this paper, a comprehensive, self-contained treatment of complex manifold theory is presented, focusing on results applicable to projective varieties, and including discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex.
Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Proceedings ArticleDOI
The complexity of theorem-proving procedures
TL;DR: It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a tautology.