This book, titled "Algorithms in Real Algebraic Geometry," is part of the "Algorithms and Computation in Mathematics" series, edited by Manuel Bronstein, Arjeh M. Cohen, Henri Cohen, David Eisenbud, and Bernd Sturmfels. It is authored by Saugata Basu, Richard Pollack, and Marie-Françoise Roy, and published by Springer-Verlag Berlin Heidelberg GmbH. The book covers a wide range of topics in real algebraic geometry, including: 1. **Algebraically Closed Fields**: Definitions, properties, Euclidean division, greatest common divisor, projection theorem for constructible sets, quantifier elimination, and the transfer principle. 2. **Real Closed Fields**: Definitions, real root counting, Descartes's Law of Signs, Budan-Fourier Theorem, Cauchy Index, sign determination, projection theorem for semi-algebraic sets, applications, Puiseux series. 3. **Semi-Algebraic Sets**: Topology, semi-algebraically connected sets, semi-algebraic germs, closed and bounded semi-algebraic sets, implicit function theorem. 4. **Algebra**: Quadratic forms, root counting, resultant and subresultant coefficients, Hilbert's Nullstellensatz, zero-dimensional systems, multivariate Hermite's quadratic form, projective space, weak Bézout's theorem. 5. **Decomposition of Semi-Algebraic Sets**: Cylindrical decomposition, semi-algebraically connected components, dimension, semi-algebraic description of cells, stratification, simplicial complexes, triangulation, Hardt's Triviality Theorem, semi-algebraic Sard's Theorem. 6. **Elements of Topology**: Simplicial homology theory, homology groups of simplicial complexes, Mayer-Vietoris theorem, chain homotopy, Euler-Poincaré characteristic. 7. **Quantitative Semi-Algebraic Geometry**: Morse theory, sum of Betti numbers, bounding Betti numbers, sum of Betti numbers of closed semi-algebraic sets. 8. **Complexity of Basic Algorithms**: Definition of complexity, linear algebra, remainder sequences, subresultants. 9. **Cauchy Index and Applications**: Cauchy index, signed remainder sequence, signed subresultant coefficients, Bezoutian, Cauchy index computation, Hankel matrices, number of complex roots with negative real part. 10. **Real Roots**: Bounds on roots, isolating real roots, sign determination, roots in a real closed field. 11. **Polynomial System Solving**: Gröbner bases, multiplication tables, univariate representation, limits of solutions, finding points in connected components, computing Euler-Poincaré characteristic. 12. **Cylindrical Decomposition Algorithm**: Computing cylindrical decomposition, decision problem, quantifier elimination, computation of stratifying familiesThis book, titled "Algorithms in Real Algebraic Geometry," is part of the "Algorithms and Computation in Mathematics" series, edited by Manuel Bronstein, Arjeh M. Cohen, Henri Cohen, David Eisenbud, and Bernd Sturmfels. It is authored by Saugata Basu, Richard Pollack, and Marie-Françoise Roy, and published by Springer-Verlag Berlin Heidelberg GmbH. The book covers a wide range of topics in real algebraic geometry, including: 1. **Algebraically Closed Fields**: Definitions, properties, Euclidean division, greatest common divisor, projection theorem for constructible sets, quantifier elimination, and the transfer principle. 2. **Real Closed Fields**: Definitions, real root counting, Descartes's Law of Signs, Budan-Fourier Theorem, Cauchy Index, sign determination, projection theorem for semi-algebraic sets, applications, Puiseux series. 3. **Semi-Algebraic Sets**: Topology, semi-algebraically connected sets, semi-algebraic germs, closed and bounded semi-algebraic sets, implicit function theorem. 4. **Algebra**: Quadratic forms, root counting, resultant and subresultant coefficients, Hilbert's Nullstellensatz, zero-dimensional systems, multivariate Hermite's quadratic form, projective space, weak Bézout's theorem. 5. **Decomposition of Semi-Algebraic Sets**: Cylindrical decomposition, semi-algebraically connected components, dimension, semi-algebraic description of cells, stratification, simplicial complexes, triangulation, Hardt's Triviality Theorem, semi-algebraic Sard's Theorem. 6. **Elements of Topology**: Simplicial homology theory, homology groups of simplicial complexes, Mayer-Vietoris theorem, chain homotopy, Euler-Poincaré characteristic. 7. **Quantitative Semi-Algebraic Geometry**: Morse theory, sum of Betti numbers, bounding Betti numbers, sum of Betti numbers of closed semi-algebraic sets. 8. **Complexity of Basic Algorithms**: Definition of complexity, linear algebra, remainder sequences, subresultants. 9. **Cauchy Index and Applications**: Cauchy index, signed remainder sequence, signed subresultant coefficients, Bezoutian, Cauchy index computation, Hankel matrices, number of complex roots with negative real part. 10. **Real Roots**: Bounds on roots, isolating real roots, sign determination, roots in a real closed field. 11. **Polynomial System Solving**: Gröbner bases, multiplication tables, univariate representation, limits of solutions, finding points in connected components, computing Euler-Poincaré characteristic. 12. **Cylindrical Decomposition Algorithm**: Computing cylindrical decomposition, decision problem, quantifier elimination, computation of stratifying families