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67
Noncommutative Elimination in Ore Algebras Proves Multivariate Identities
 J. SYMBOLIC COMPUT
, 1996
"... ... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation. ..."
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Cited by 90 (9 self)
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... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation.
Polynomial interpolation in several variables
 J. Algebraic Geom
, 1995
"... This is a survey of the main results on multivariate polynomial interpolation in the last twentyfive years, a period of time when the subject experienced its most rapid development. The problem is considered from two different points of view: the construction of data points which allow unique inter ..."
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Cited by 73 (0 self)
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This is a survey of the main results on multivariate polynomial interpolation in the last twentyfive years, a period of time when the subject experienced its most rapid development. The problem is considered from two different points of view: the construction of data points which allow unique interpolation for given interpolation spaces as well as the converse. In addition, one section is devoted to error formulas and another to connections with computer algebra. An extensive list of references is also included.
Polynomial Interpolation in Several Variables
, 1999
"... this paper we want to describe some recent developments in polynomial interpolation, especially those which lead to the construction Partially supported by DGES Spain, PB 960730 Partially supported by DGES Spain, PB 960730 and Programa Europa CAIDGA, Zaragoza, Spain 2 M. Gasca and T. Sauer / Po ..."
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Cited by 40 (2 self)
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this paper we want to describe some recent developments in polynomial interpolation, especially those which lead to the construction Partially supported by DGES Spain, PB 960730 Partially supported by DGES Spain, PB 960730 and Programa Europa CAIDGA, Zaragoza, Spain 2 M. Gasca and T. Sauer / Polynomial interpolation of the interpolating polynomial, rather than verification of its mere existence
Polar varieties and efficient real elimination
 MATH. Z
, 2001
"... Let S0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1,..., fp. This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S0. For this purpose we exhibit explicit polynomial equations th ..."
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Cited by 29 (12 self)
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Let S0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1,..., fp. This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S0. For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S0. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f1,..., fp and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f1,..., fp.
Polar varieties, real equation solving and data structures: The hypersurface case
 J. COMPLEXITY
, 1997
"... In this paper we apply for the rst time a new method for multivariate equation solving which was developed in [18], [19], [20] for complex root determination to the real case. Our main result concerns the problem of nding at least one representative point foreachconnected component of a real compac ..."
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Cited by 23 (11 self)
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In this paper we apply for the rst time a new method for multivariate equation solving which was developed in [18], [19], [20] for complex root determination to the real case. Our main result concerns the problem of nding at least one representative point foreachconnected component of a real compact and smooth hypersurface. The basic algorithm of [18], [19], [20] yields a new method for symbolically solving zerodimensional polynomial equation systems over the complex numbers. feature of central importance of this algorithm is the use of a problem{adapted data type represented by the data structures arithmetic network and straightline program (arithmetic circuit). The algorithm nds the complex solutions of any a ne zerodimensional equation system in nonuniform sequential time that is polynomial in the length of the input (given in straight{line program representation) and an adequately de ned geometric degree of the equation system. Replacing the notion of geometric degree of the given polynomial equation system by a suitably de ned real (or complex) degree of certain polar varieties associated to
Solving and factoring boundary problems for linear ordinary differential equations in differential algebras
 Journal of Symbolic Computation
"... We present a new approach for expressing and solving boundary problems for linear ordinary differential equations in the language of differential algebras. Starting from an algebra with a derivation and integration operator, we construct a ring of linear integrodifferential operators that is expres ..."
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Cited by 16 (12 self)
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We present a new approach for expressing and solving boundary problems for linear ordinary differential equations in the language of differential algebras. Starting from an algebra with a derivation and integration operator, we construct a ring of linear integrodifferential operators that is expressive enough for specifying regular boundary problems with arbitrary Stieltjes boundary conditions as well as their solution operators. Based on these structures, we define a new multiplication on regular boundary problems in such a way that the resulting Green’s operator is the reverse composition of the constituent Green’s operators. We provide also a method for lifting any factorization of the underlying differential operator to the level of boundary problems. Since this method only needs the computation of initial value problems, it can be used as an effective alternative for computing Green’s operators in case one knows how to factor the given differential operators.
A Superexponential Lower Bound for Gröbner Bases and ChurchRosser Commutative Thue Systems
, 1986
"... The complexity of the normal form algorithms which transform a given polynomial ideal basis into a Gröbner basis or a given commutative Thue system into a ChurchRosser system is presently unknown. In this paper we derive a doubleexponential lower bound (22") for the production length and cardinali ..."
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Cited by 16 (0 self)
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The complexity of the normal form algorithms which transform a given polynomial ideal basis into a Gröbner basis or a given commutative Thue system into a ChurchRosser system is presently unknown. In this paper we derive a doubleexponential lower bound (22") for the production length and cardinality of ChurchRosser commutative Thue systems, and the degree and cardinality of Gröbner bases.
Computer algebra and algebraic geometry  achievements and perspectives
 J. SYMBOLIC COMPUT
, 2000
"... In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One of the aims of this paper is to show, by means of example ..."
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Cited by 12 (1 self)
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In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One of the aims of this paper is to show, by means of examples, the usefulness of computer algebra to mathematical research. Computer algebra itself is a highly diversified discipline with applications to various areas of mathematics; many of these may be found in numerous research papers, proceedings or textbooks (cf. Buchberger and Winkler, 1998; Cohen et al., 1999; Matzat et al., 1998; ISSAC, 1988–1998). Here, I concentrate mainly on Gröbner bases and leave aside many other topics of computer algebra (cf. Davenport et al., 1988; Von zur Gathen and Gerhard, 1999; Grabmeier et al., 2000). In particular, I do not mention (multivariate) polynomial factorization, another major and important tool in computational algebraic geometry. Gröbner bases were introduced originally by Buchberger as a computational tool for testing solvability of a system of polynomial equations, to count the number of solutions (with multiplicities) if this number is finite and, more algebraically, to compute in the quotient ring modulo the given polynomials. Since then, Gröbner bases have become the major computational tool, not only in algebraic geometry. The importance of Gröbner bases for mathematical research in algebraic geometry is obvious and nowadays their use needs hardly any justification. Indeed, chapters on Gröbner bases and Buchberger’s algorithm (Buchberger, 1965) have been incorporated in many new textbooks on algebraic geometry such as the books of Cox et al. (1992, 1998) or the recent books of Eisenbud (1995) and Vasconcelos (1998), not to mention textbooks which are devoted exclusively to Gröbner bases, such as Adams and Loustaunou (1994),
Computing Combinatorial Decompositions Of Rings
, 1991
"... This article deals with a topic on the borderline of commutative ring theory, computer algebra and combinatorics. We study canonical decompositions of commutative Noetherian rings. These techniques are based on earlier results of Rees [Ree], Stanley [St2], and Baclawski & Garsia [BGa], and they gene ..."
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Cited by 11 (1 self)
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This article deals with a topic on the borderline of commutative ring theory, computer algebra and combinatorics. We study canonical decompositions of commutative Noetherian rings. These techniques are based on earlier results of Rees [Ree], Stanley [St2], and Baclawski & Garsia [BGa], and they generalize the wellknown Hironaka decomposition of CohenMacaulay rings. Here it is our main objective to give explicit algorithms for computing these decompositions.