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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
Citations
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Can one design a geometry engine
TL;DR: It is shown here using elementary model theoretic tools that the universal first order consequences of any geometric theory T of Pappian planes which is consistent with the analytic geometry of the reals is decidable.
Posted Content
Topics in vacuum decay (Ph.D Thesis)
TL;DR: In this paper, the authors studied the stability of vacua in a landscape of potentials and the possible instability in a large dimension of fields with different symmetry than the generically used O(4) symmetry.
Journal ArticleDOI
A sequence of polynomials with optimal condition number
TL;DR: In this paper, an explicit sequence of univariate polynomials of arbitrary degree with optimal condition number was found. But this problem was not solved by the authors of this paper.
Journal ArticleDOI
Uncomputably large integral points on algebraic plane curves
TL;DR: In this paper, it was shown that the decidability of an amplification of Hilbert's Tenth Problem in three variables implies the existence of uncomputably large integral points on certain algebraic curves.
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Derandomization and absolute reconstruction for sums of powers of linear forms
Pascal Koiran,Mateusz Skomra +1 more
TL;DR: An algorithm for the following problem: given a homogeneous polynomial of degree 3, decide whether it can be written as a sum of cubes of linearly independent linear forms with complex coefficients to solve the decomposition of multivariate polynomials as sums of powers of linear forms.
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.