Basu pollack roy 2006 algorithms real algebraic geometry

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theorem_1_22: Basu, Pollack and Roy's projection theorem for constructible sets: over an algebraically closed field C, the projection to C^k of a constructible set in C^{k+1} defined by polynomials with coefficients in a subring D is a constructible set defined by polynomials with coefficients in D.

theorem_1_23: Basu, Pollack and Roy's quantifier elimination over an algebraically closed field C: every formula of the language of fields with free variables Y_1, ..., Y_l and coefficients in a subring D of C is C-equivalent to a quantifier-free formula in the same variables with coefficients in D.

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Saugata Basu, Richard Pollack, and Marie-Françoise Roy, Algorithms in Real Algebraic Geometry, 2nd ed., Springer, 2006, Algorithms and Computation in Mathematics 10, X+662 pages, DOI 10.1007/3-540-33099-2.

The copy read for this card is a 672-page scan including front matter, whose title page displays the authors and title but no edition or year. Springer metadata identifies the second edition and its X+662 book pages. This correspondence establishes a catalog match, not publisher-facsimile byte identity. That scan prints a Springer title page but a blank copyright page and no notice, so it may be the authors' posted version rather than the publisher's facsimile; the publisher's book page states "© Springer-Verlag Berlin Heidelberg 2006" and carries no open access statement (https://link.springer.com/book/10.1007/3-540-33099-2), every other right reserved.

The page numbers cited on this card and its result pages are those printed in the copy read. That copy numbers its pages continuously from the title page (Introduction p. 9, Chapter 1 p. 19), and its introduction points to an errata list for the second edition (p. 17), so its pagination need not match the Springer print.

Chapter 1 works over an algebraically closed field $\mathrm C$ (p. 19), and Section 1.2 fixes a subring $D$ of $\mathrm C$ with quotient field $K$ (p. 22); the field $\mathbb C$ of complex numbers is the typical example, not the standing hypothesis. Constructible sets are the smallest family containing the algebraic sets and closed under complement, finite union and finite intersection (p. 20). Theorem 1.22 (p. 33) states that a constructible set in $\mathrm C^{k+1}$ defined by polynomials with coefficients in $D$ has a projection to $\mathrm C^k$ that is constructible and defined by polynomials with coefficients in $D$. Theorem 1.23 on the same page states quantifier elimination over the algebraically closed field $\mathrm C$: for a subring $D$ of $\mathrm C$, every formula $\Phi(Y_1,\ldots,Y_\ell)$ of the language of fields whose free variables are $\{Y_1,\ldots,Y_\ell\}$ and whose coefficients lie in $D$ is $\mathrm C$-equivalent to some quantifier-free formula $\Psi(Y_1,\ldots,Y_\ell)$ with coefficients in $D$. The book derives from it the Lefschetz principle (Theorem 1.26, p. 34).

Read status: claims checked for Theorems 1.22 and 1.23 and the definitions of Section 1.1 they use, read clause by clause on the page images; the induction proving Theorem 1.23 was followed, and the proof of Theorem 1.22 was read for structure only. Nothing here is independently reviewed. The rest of the book was not read for this card.

Bears on. None. The book states no relation to any numbered Erdős problem.

Results.

• Theorem 1.22 (p. 33): the projection of a constructible set is constructible, with coefficients kept in $D$.
• Theorem 1.23 (p. 33): quantifier elimination over algebraically closed fields, with coefficients kept in $D$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.