Abstract
The preceding just touches on some of the highlights of the accomplishments and unsolved problems in computer algebra. A really comprehensive survey would be much too long for the space available here. I close with the following quote, which has been attributed to Albert Einstein and helps, perhaps, to keep a proper prospective on our work: "The symbolic representation of abstract entities is doomed to its rightful place of relative insignificance in a world in which flowers and beautiful women abound."
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5. References
D.S. Arnon, Algorithms for the geometry of semi-algebraic set, Ph.D. Thesis, Tech. Report No. 436, Computer Sciences Dept., Univ. of Wisconsin (1981).
___, Topologically reliable display of algebraic curves, Proc. of SIGGRAPH '83 ACM, New York (1983), 219–227.
___, On mechanical quantifier elimination for elementary algebra and geometry: solution of a nontrivial problem, these Proceedings.
___, G.E. Collins and S. McCallum, Cylindrical algebraic decomposition I: the basic algorithm, SIAM J. on Comput. 13, 4 (1984), 865–877.
J. Ax, On Schanuel's conjectures, Annals of Math. 93 (1971), 252–268.
E.T. Bell, Men of Mathematics, Simon and Schuster, New York (1937).
E.R. Berlekamp, Factoring polynomials over finite fields, Bell Sys. Tech. J. 46, (1967), 1853–1859.
W. Böge, Decision procedures and quantifier elimination for elementary real algebra and parametric polynomial nonlinear optimization, preliminary manuscript (1980).
W.S. Brown, On Euclid's algorithm and the computation of polynomial greatest common divisors, J. ACM 18, 4 (Oct 1971), 478–504.
___, On the subresultant PRS algorithm, in [Jenk76], 271.
___, The subresultant PRS algorithm, ACM TOMS 4, (1978), 237–249.
___, J.P. Hyde and B.A. Tague, The ALPAK system for non-numerical algebra on a digital computer II: rational functions of several variables and truncated power series with rational function coefficients, Bell Sys. Tech. J. 43 (March 1964), 785–804.
___ and J.F. Traub, On Euclid's algorithm and the theory of subresultants, J. ACM 18, 4 (vol 1971), 505–514.
B. Buchberger, An algorithm for finding a basis for the residue class ring of a zerodimensional polynomial ideal (German), Ph.D. Thesis Math. Inst., Univ. of Innsbruck, Austria (1965) and Aequationes Math 4, 3 (1970), 374–383.
___, A theoretical basis for the reduction of polynomials to canonical form, ACM SIGSAM Bull. 10, 3 (1976), 19–29.
___, Some properties of Gröbner bases for polynomial ideals, ACM SIGSAM Bull. 10, 4 (1976), 19–24.
___, A criterion for detecting unnecessary reductions in the construction of Gröbner bases, in [Ng79], 3–21.
___, A note on the complexity of constructing Gröbner bases, in [vHul83], 137–145.
___, A critical-pair/completion algorithm for finitely generated ideals in rings, Proc. of the Conf. Rekursive Kombinatorik, Springer-Verlag LNCS (to appear).
___, CAMP: a teaching project in symbolic computation at the University of Linz, SIGSAM Bull. 18, 4(Nov 1984),8–9.
___, Gröbner bases: an algorithmic method in polynomial ideal theory, in Recent Trends in Multidimensional Systems Theory, N.K. Bose (ed.), D. Riedel Publ. Co. (1985), to appear.
___, G.E. Collins and R. Loos (eds), Computer Algebra: Symbolic and Algebraic Computation, 2nd ed. (1983), Springer-Verlag, Vienna.
___ and R. Loos, Algebraic simplification, in [BCL83], 11–43.
J.A. Campbell, J.G. Kent and R.J. Moore, Experiments with a symbolic programming system for complex analysis, BIT 16 (1976), 241–256.
G.W. Cherry, Algorithms for integrating elementary functions in terms of logarithmic integrals and error functions, Ph.D. Thesis, Univ. of Delaware (August 1983).
___, Integration in finite terms with special functions: the logarithmic integral, SIAM J. on Comput. (to appear).
___, Integration in finite terms with special functions: the error function, J. Symbolic Computation, (to appear).
P.J. Cohen, Decision procedures for real and p-adic fields, Comm. Pure and Applied Math. 22 (1969), 131–151.
G.E. Collins, PM, a system for polynomial manipulation, Comm. ACM 9, 8 (1966), 578–589.
___, Subresultants and reduced polynomial remainder sequences, J. ACM 14 (Jan 1967), 128–142.
___, The SAC-2 polynomial GCD and resultant system, U. of Wisconsin Comp. Sci. Tech. Report #145 (Feb. 1972).
___, The computing time of the Euclidean algorithm, SIAM J. on Comput. 3, 1 (1974), 1–10.
___, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Proc. of Second GI Conf. on Automata Theory and Formal Languages, Springer-Verlag LNCS 33 (1975), 134–183.
___, Factoring univariate integral polynomials in polynomial average time, in [Ng79], 317–329.
___, Quantifier elimination for real closed fields: a guide to the literature, in [BCL83], 79–81.
___ and A.G. Akritas, Polynomial real root isolation using Descartes rule of signs, in [Jenk76], 272–275.
___ and R. Loos, Real zeros of polynomials, in [BCL83], 83–94.
J.H. Davenport, On the Integration of Algebraic Functions, Springer-Verlag LNCS 102, (1981).
J.D. Dixon, The number of steps in the Euclidean algorithm, J. Number Theory 2 (1970), 414–422.
E. Engeler, Scientific computation: the integration of symbolic, numeric and graphic computation, these Proceedings (1985).
M.J. Fischer and M.O. Rabin, Super exponential complexity of Presburger arithmetic, MIT MAC Tech Memo 43 (1974).
J. Fitch, Proc. 1984 European Symp. on Symbolic and Algebraic Manipulation, Springer-Verlag LNCS 174 (1984).
R.W. Floyd (ed.), Proceedings of the ACM symposium on symbolic and algebraic manipulation, Comm. ACM 9, 8 (Aug 1966), 547–643.
R. Gebauer and H. Kredel, An algorithm for constructing Gröbner bases of polynomial ideals, SIGSAM Bull. 18, 1 (1984), 9.
R.W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. 75, 1 (1978), 40–42.
S.J. Harrington, A new symbolic integration system for Reduce, Computer J. 22 (1979),127–131.
H.A. Heilbronn, On the average length of a class of finite continued fractions, in Number Theory and Analysis, Paul Turán (ed.), Plenum, New York (1969), 87–96.
L.E. Heindel, Integer arithmetic algorithms for polynomial real zero determination, J. ACM 18, 4 (1971), 533–548.
G. Hermann, The question of finitely many steps in polynomial ideal theory (German), Math. Annal. 95 (1926), 736–788.
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II, Annals of Math. 79 (1964), 109–326.
G. Huet and D.C. Oppen, Equations and rewrite rules, in Formal Language Theory: Perspectives and Open Problems, R.V. Book (ed.), Academic Press, New York (1980), 349–405.
R.D. Jenks (ed.), 1976 ACM Symposium on Symbolic and Algebraic Computation, ACM, New York, NY (August 1976).
D.E. Jordon, R.Y. Kain and L.C. Clapp, Symbolic factoring of polynomials in several variables, Comm. ACM 8, 9 (August 1966), 638–643.
S.C. Johnson, Tricks for improving Kronecker's polynomial factoring algorithm, Bell Telephone Labs Tech. Report.
E. L. Kaltofen, Factorization of polynomials, in [BCL83], 95–113.
___, On the complexity of factoring polynomials with integer coefficients, Ph.D. Thesis, Rensselaer Polytechnic Inst. (1982).
___, A polynomial-time reduction from bivariate to univariate integral polynomial factorization, Proc. 23rd Symposium on Foundations of Computer Science, IEEE (1982), 57–64.
___, A polynomial reduction from multivariate to bivariate polynomial factorization, Proc. 14th Annual ACM Symp. on the Theory of Comput. (1982), 261–266.
A. Kandri-Rody and D. Kapur, On the relationship between Buchberger's Gröbner basis algorithm and the Knuth-Bendix completion procedure, TIS Report No. 83CRD286, General Electric R & D Ctr. (1983).
___, Algorithms for computing Gröbner bases of polynomial ideals over various Euclidean rings, in [Fitc84], 195–206.
___, Computing the Gröbner basis of polynomial ideals over integers, Proc. of 3rd Macsyma Users' Conf., Schenectady, NY (1984).
M. Karr, Summation in finite terms, J. ACM 28, 2 (1981), 305–350.
D.E. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, Addison-Wesley, Reading, Mass. (1969).
J.J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, manuscript (1979).
S. Landau, Factoring polynomials over algebraic number fields, SIAM J. on Comput. 14, 1 (1985), 184–195.
___ and G. Miller, Solvability by radicals is in polynomial time, Proc. 15th Annual ACM Symp. on Theory of Comp. (1983), 140–151.
D. Lazard, Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations, in [vHul83], 146–156.
M. Lauer, Canonical representatives for residue classes of a polynomial ideal, in [Jenk76], 339–345.
___, Computing in homomorphic images, in [BCL83], 139–168.
A.K. Lenstra, H.W. Lenstra and L. Lovász, Factoring polynomials with rational coefficients, Math. Annal. 261 (1982), 515–534.
V.E. Lewis (ed.), Proc. 1977 Macsyma Users' Conf., MIT, Cambridge, MA (1977).
R. Loos, Toward a formal implementation of computer algebra, Proc. 1974 European Symposium on Symbolic and Algebraic Manipulation, ACM SIGSAM Bull 8, 3 (1974), 9–16.
___, Generalized polynomial remainder sequences, in [BCL83], 115–137.
E.W. Mayr and A.R. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Advances in Math 46 (1982), 305–329.
F. Mora and H.M. Möller, The computation of the Hilbert function, in [vHul83], 157–167.
J. Moses, Symbolic integration, Ph.D. Thesis, M.I.T. (1967), also Project MAC Tech. Report TR-47.
___, The integration of a class of special functions with the Risch algorithm, SIGSAM Bull., No. 13 (December 1969), 13–27.
J. Moses and D.Y.Y. Yun, The EZGCD algorithm, Proc. ACM National Conf., (Aug 1973), 159–166.
F. Müller, Ein exakter Algorithmus zur nichtlinearen Optimierung für beliebige Polynome mit mehreren Veranderlichen, Verlag Anton Hain, Meisenheim am Glan (1978).
D.R. Musser, Algorithms for polynomial factorization, Ph.D. Thesis, Dept. of Computer Sciences, Univ. of Wisconsin (1971).
___, Multivariate polynomial factorization, J. ACM 22, (1976), 291–308.
E.W. Ng (ed.), Proc. 1979 European Symposium on Symbolic and Algebraic Manipulation, Spring-Verlag LNCS 72 (1979).
J.R. Pinkert, An exact method for finding the roots of a complex polynomial, ACM TOMS 2 (1976), 351–363.
J.W. Porter, Mathematika 22 (1975), 20–28.
M. Pohst and D.Y.Y. Yun. On solving systems of algebraic equations via ideal bases and elimination theory, in [Wang81], 206–211.
D. Richardson, Some unsolvable problems involving elementary functions of a real variable, J. Symb. Logic 33 (1968), 511–520.
R.H. Risch, On the integration of elementary functions which are built up using algebraic operations, System Development Corp. Tech. Report SP-2801/002/00 (1968).
___, The problem of integration in finite terms, Trans. AMS 139, (May 1969), 167–189.
___, Further results on elementary functions, IBM Research Tech Report RC 2402 (March 1969).
___, The solution of the problem of integration in finite terms, Bull. AMS 76, (1970), 605–608.
___, Algebraic properties of the elementary functions of analysis, Amer. J. Math. 101 (1979), 743–759.
H. Rolletschek, The Euclidean algorithm for Gaussian integers, in [vHu183], 12–23.
M. Rosenlicht, On Liouville's theory of elementary functions, Pacific J. of Math. 65, 2 (1976), 485–492.
M. Rothstein, Aspects of symbolic integration and simplification of exponential and primitive functions, Ph.D. Thesis, Univ. of Wisconsin (1976).
B.D. Saunders, An implementation of Kovacic's algorithm for solving second order linear homogeneous differential equations, in [Wang81], 105–108.
S. Schaller, Algorithmic aspects of polynomial residue class rings, Ph.D. Thesis., Dept. of Comp. Sci., Univ. of Wisconsin (1979).
A. Seidenberg, A new decision method for elementary algebra, Ann. of Math. 60 (1954), 365–374.
M.F. Singer, Liouvillian solutions of nth order homogeneous linear differential equations, Amer. J. Math. 103, 4 (1980), 661–682.
___, Solving homogeneous linear differential equations in terms of second order linear differential equations, preprint.
___, B.D. Saunders and B.F. Caviness, An extension of Liouville's theorem on integration in finite terms, SIAM J. on Comput. (to appear).
J.R. Slagle, A heuristic program that solves symbolic integration problems in freshman calculus, symbolic automatic integrator (SAINT), Ph.D. Thesis, Mass. Inst. of Tech. (1961).
D. Spear, A constructive approach to commutative ring theory, in [Lewi77], 369–376.
G. Szekeres, A canonical basis for the ideals of a polynomial domain, Am. Math. Monthly 59, 6 (1952), 279–386.
A. Tarski, A decision method for elementary algebra and geometry, 2nd revised ed., Univ. of California Press (1951).
B.M. Trager, Algebraic factoring and rational function integration, in [Jenk76], 219–226.
___, Integration of simple radical extensions, in [Ng79], 408–414.
B. L. van der Waerden, Modern Algebra, Vol. I, Frederick Ungar, New York (1953).
J.A. van Hulzen, Computer Algebra: Proceedings of Eurocal '83, Springer-Verlag LNCS 162, (1983).
J. von zur Gathen, Factoring sparse multivariate polynomials, Proc. of the 24th Annual IEEE Symp. on the Foun. of Comp. Sci., (1983), 172–179.
P.S. Wang, Preserving sparseness in multivariate polynomial factorization, in [Lewi77], 55–61.
___, An improved multivariate polynomial factoring algorithm, Math. Comp. 32 (1978), 1215–1231.
___, Analysis of the p-adic construction of multivariate correction coefficients in polynomial factorization: iteration vs. recursion, in [Ng79], 291–300.
___ (ed.) Proc. 1981 ACM Symp. on Symbolic and Algebraic Computation (1981), ACM, New York.
___ and L.P. Rothschild, Factoring multivariate polynomials over the integers, Math. Comp. 29 (1975), 935–950.
L.H. Williams, Algebra of polynomials in several variables for a digital computer, J. ACM, 9, 1 (Jan 1962), 29–40.
F. Winkler, An algorithm for constructing detaching bases in the ring of polynomials over a field, in [vHul83], 168–179.
_____, On the complexity of the Gröbner-basis algorithm over K[x,y,z], in [Fitc84], 184–194.
___, B. Buchberger, F. Lichtenberger and H. Rolletschek, An algorithm for constructing canonical bases of polynomial ideals, ACM TOMS 11, 1 (1985), 66–78.
D.Y.Y. Yun, The Hensel lemma in algebraic manipulation, PhD Thesis, M.I.T., MAC-TR-138 (Nov 1974).
H. Zassenhaus, On Hensel factorization I, J. Num. 1 (1969), 287–292.
G. Zacharias, Generalized Gröbner bases in commutative polynomial rings, Bachelor Thesis, MIT (1978).
R.E. Zippel, Probabilistic algorithms for sparse polynomials, Ph.D. Thesis, MIT (1979).
_____, Probabilistic algorithms for sparse polynomials, in [Ng79], 216–226.
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Caviness, B.F. (1985). Computer algebra: Past and future. In: Buchberger, B. (eds) EUROCAL '85. EUROCAL 1985. Lecture Notes in Computer Science, vol 203. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-15983-5_1
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