George E. Collins

Cylindrical algebraic decomposition I: the basic algorithm

Given a set of r-variate integral polynomials, a cylindrical algebraic decomposition (cad) of euclidean r-space E r partitions E r into connected subsets compatible with the zeros of the polynomials. By “compatible with the zeros of the polynomials” we mean that on each subset of E r , each of the polynomials either vanishes everywhere or nowhere. For example, consider the bivariate polynomial

Computer algebra symbolic and algebraic computation

{y^4} - 2{y^3} + {y^2} - 3{x^2}y + 2{x^4}.

Real Zeros of Polynomials

Let @@@@ be an integral domain, P(@@@@) the integral domain of polynomials over @@@@. Let <italic>P</italic>, <italic>Q</italic> ∈ P(@@@@) with <italic>m</italic> @@@@ deg (<italic>P</italic>) ≥ <italic>n</italic> = deg (<italic>Q</italic>) > 0. Let <italic>M</italic> be the matrix whose determinant defines the resultant of <italic>P</italic> and <italic>Q</italic>. Let <italic>M<subscrpt>ij</subscrpt></italic> be the submatrix of <italic>M</italic> obtained by deleting the last <italic>j</italic> rows of <italic>P</italic> coefficients, the last <italic>j</italic> rows of <italic>Q</italic> coefficients and the last 2<italic>j</italic>+1 columns, excepting column <italic>m</italic> — <italic>n</italic> — <italic>i</italic> — <italic>j</italic> (0 ≤ <italic>i</italic> ≤ <italic>j</italic> < <italic>n</italic>). The polynomial <italic>R<subscrpt>j</subscrpt></italic>(<italic>x</italic>) = ∑<supscrpt><italic>i</italic></supscrpt><subscrpt><italic>i</italic>=0</subscrpt> det (<italic>M<subscrpt>ij</subscrpt></italic>)<italic>x<supscrpt>i</supscrpt></italic> is the <italic>j-t subresultant</italic> of <italic>P</italic> and <italic>Q</italic>, <italic>R<subscrpt>0</subscrpt></italic> being the resultant. If <italic>b</italic> = £(<italic>Q</italic>), the leading coefficient of <italic>Q</italic>, then exist uniquely <italic>R</italic>, <italic>S</italic> ∈ P(@@@@) such that <italic>b</italic><supscrpt><italic>m-n</italic>+1</supscrpt> <italic>P</italic> = <italic>QS</italic> + <italic>R</italic> with deg (<italic>R</italic>) < <italic>n</italic>; define R(<italic>P</italic>, <italic>Q</italic>) = <italic>R</italic>. Define <italic>P<subscrpt>i</subscrpt></italic> ∈ P(<italic>F</italic>), <italic>F</italic> the quotient field of @@@@, inductively: <italic>P</italic><subscrpt>1</subscrpt> = <italic>P</italic>, <italic>P</italic><subscrpt>2</subscrpt> = <italic>Q</italic>, <italic>P</italic><subscrpt>3</subscrpt> = R<italic>P</italic><subscrpt>1</subscrpt>, <italic>P</italic><subscrpt>2</subscrpt> <italic>P</italic><subscrpt><italic>i</italic>-2</subscrpt> = R(<italic>P<subscrpt>i</subscrpt></italic>, <italic>P</italic><subscrpt><italic>i</italic>+1</subscrpt>)/<italic>c</italic><supscrpt>δ<subscrpt><italic>i</italic>-1</subscrpt>+1</supscrpt><subscrpt><italic>i</italic></subscrpt> for <italic>i</italic> ≥ <italic>2</italic> and <italic>n</italic><subscrpt><italic>i</italic>+1</subscrpt> > 0, where <italic>c</italic><subscrpt><italic>i</italic></subscrpt> = £(<italic>P<subscrpt>i</subscrpt></italic>), <italic>n<subscrpt>i</subscrpt></italic> = deg (<italic>P<subscrpt>i</subscrpt></italic>) and δ<subscrpt><italic>i</italic></subscrpt> = <italic>n<subscrpt>i</subscrpt></italic> — <italic>n</italic><subscrpt><italic>i</italic>+1</subscrpt>. <italic>P</italic><subscrpt>1</subscrpt>, <italic>P</italic><subscrpt>2</subscrpt>, …, <italic>P<subscrpt>k</subscrpt></italic>, for <italic>k</italic> ≥ 3, is called a <italic>reduced polynomial remainder sequence</italic>. Some of the main results are: (1) <italic>P<subscrpt>i</subscrpt></italic> ∈ P(@@@@) for 1 ≤ <italic>i</italic> ≤ <italic>k</italic>; (2) <italic>P<subscrpt>k</subscrpt></italic> = ± <italic>A<subscrpt>k</subscrpt>R</italic><subscrpt><italic>n</italic><subscrpt><italic>k</italic>-1</subscrpt>-1</subscrpt>, when <italic>A<subscrpt>k</subscrpt></italic> = &Pgr;<supscrpt><italic>k</italic>-2</supscrpt><subscrpt><italic>i</italic>-2</subscrpt><italic>c</italic><supscrpt>δ<subscrpt><italic>i</italic>-1</subscrpt>(δ<italic>i</italic>-1)</supscrpt><subscrpt><italic>i</italic></subscrpt>; (3) <italic>c</italic><supscrpt>δ<subscrpt><italic>k</italic>-1</subscrpt>-1</supscrpt><subscrpt><italic>k</italic></subscrpt> <italic>P<subscrpt>k</subscrpt></italic> = ±<italic>A</italic><subscrpt><italic>k</italic>+1</subscrpt><italic>R</italic><subscrpt><italic>n</italic><subscrpt><italic>k</italic></subscrpt></subscrpt>; (4) <italic>R<subscrpt>j</subscrpt></italic> = 0 for <italic>n<subscrpt>k</subscrpt></italic> < <italic>j</italic> < <italic>n</italic><subscrpt><italic>k</italic>-1</subscrpt> — 1. Taking @@@@ to be the integers <italic>I</italic>, or P<supscrpt><italic>r</italic></supscrpt>(<italic>I</italic>), these results provide new algorithms for computing resultant or greatest common divisors of univariate or multivariate polynomials. Theoretical analysis and extensive testing on a high-speed computer show the new g.c.d. algorithm to be faster than known algorithms by a large factor. When applied to bivariate polynomials, for example this factor grows rapidly with the degree and exceeds 100 in practical cases.

Cylindrical algebraic decomposition II: an adjacency algorithm for the plane

Computer algebra is an alternative and complement to numerical mathematics. Its importance is steadily increasing. This volume is the first systematic and complete treatment of computer algebra. It presents the basic problems of computer algebra and the best algorithms now known for their solution with their mathematical foundations, and complete references to the original literature. The volume follows a top-down structure proceeding from very high-level problems which will be well-motivated for most readers to problems whose solution is needed for solving the problems at the higher level. The volume is written as a supplementary text for a traditional algebra course or for a general algorithms course. It also provides the basis for an independent computer algebra course.

The SAC-1 system: An introduction and survey

An important property of the Newell Shaw-Simon scheme for computer storage of lists is that data having multiple occurrences need not be stored at more than one place in the computer. That is, lists may be “overlapped.” Unfortunately, overlapping poses a problem for subsequent erasure. Given a list that is no longer needed, it is desired to erase just those parts that do not overlap other lists. In LISP, McCarthy employs an elegant but inefficient solution to the problem. The present paper describes a general method which enables efficient erasure. The method employs interspersed reference counts to describe the extent of the overlapping.

Polynomial real root isolation by differentiation

An efficient algorithm is presented for the exact calculation of resultants of multivariate polynomials with integer coefficients. The algorithm applies modular homomorphisms and the Chinese remainder theorem, evaluation homomorphisms and interpolation, in reducing the problem to resultant calculation for univariate polynomials over GF(p), whereupon a polynomial remainder sequence algorithm is used. The c o m p u t i n g t i m e of t h e a l g o r i t h m is a n a l y z e d t h e o r e t i c a l l y a s a f u n c t i o n of t he d e g r e e s a n d c o e f f i c i e n t s i z e s of i t s i n p u t s . As a v e r y s p e c i a l c a s e , i t i s s h o w n t h a t w h e n a l l d e g r e e s a r e e q u a l a n d t h e c o e f f i c i e n t s i z e i s f i x e d , i t s c o m p u t i n g t ime i s a p p r o x i m a t e l y p r o p o r t i o n a l to X 2 r+ l , w h e r e X i s t h e c o m m o n d e g r e e a n d r i s t he n u m b e r of v a r i a b l e s . Empirically observed computing times of the algorithm are tabulated for a large number of examples, and other algorithms are compared. Potential application of the algorithm to the solution of systems of polynomial equations is briefly discussed. I . I n t r o d u c t i o n Let R be a commutative ring with an identity element and let A and B be polynomials of positive d e g r e e w i t h c o e f f i c i e n t s in R. I f A(x) = Z i = o a m xi a n d B(x) = Z i = o b i x n i w h e r e deg(A) = m a n d deg(B) = n , t h e S y l . v e s t e r m a t r i x of A a n d B is t he m + n b y m + n matrix

PM, a system for polynomial manipulation

Let A be a polynomial over Z, Q or Q(α) where α is a real algebraic number. The problem is to compute a sequence of disjoint intervals with rational endpoints, each containing exactly one real zero of A and together containing all real zeros of A. We describe an algorithm due to Kronecker based on the minimum root Separation, Sturm’s algorithm, an algorithm based on Rolle’s theorem due to Collins and Loos and the modified Uspensky algorithm due to Collins and Aritas. For the last algorithm a recursive version with correctness proof is given which appears in print for the first time.