D. Bayer and M. Stillman. Macaulay: a computer algebra system for algebraic geometry. available by anonymous ftp from ftp.math.harvard.edu.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....if depth V (R=J) 1, which means the ideal of V contains some non zero divisor mod J . Note, by the way, that the proof of Proposition 6.1 works equally well with more than two sets of variables. Some explorations we have done for small values of n using the computer algebra system MACAULAY [1] suggest the following conjecture. Conjecture 6.2. If J denotes the ideal generated by the S n alternants in C[x; y; z] for any number of sets of n variables, and V is the locus where all the points (x i ; y i ; z i ) x j ; y j ; z j ) coincide, then x 1 Gamma x 2 ; x ....

D. Bayer and M. Stillman, MACAULAY: A computer algebra system for algebraic geometry, version 3.0, Public domain computer program distributed by Harvard University (1989).

....kan sml, vhich has one line commands for many of the algorithms and applications described here and vhich is available at http: www.kobe u. ac. jp KAN. Similarly, together vith Anton Leykin and Mike Stillman, ve have implemented a package for D modules for the computer algebra system lacaulay 2 [23]. One of the nice features of Macaulay 2 is that it has a top level programming environment vhich makes the system flexible for applications. Our package vill shortly be included in the Macaulay 2 distribution and is currently available at httP: www raath berkeley edu htsai Dradules htral We ....

Grayson, D., Stillman, M. (1999): Macaulay 2: a computer algebra system for algebraic geometry, Version 0.8.56, http://www. math. uiuc. edu/biacaulay2.

....polynomials in a andb linking variables, should be empty. The same argument can be made for the case of analytic constraints. In the case of polynomial constraints, it can be computationally checked whether K ab (x) has full rank everywhere. Some symbolic computer algebra systems, e.g. Macaulay [1] and Singular [4] are capable of computing the projection of an ideal onto a subring. 5 Computational Results 5.1 Obtaining Two Distinct Decompositions Recall that the functional dependence table (FDT) is a Boolean matrix representing the dependence of design functions on variables. The (i, ....

D. Bayer and M. Stillman. Macaulay: A computer algebra system for algebraic geometry, 1995. Available by anonymous ftp from ftp.math.harvard.edu.

....transform the given equations into a special form called Grobner basis, 23, 24, 25, 26] from which the solutions can be read off fairly easily. Algorithms for performing Grobner bases calculations have been implemented in many popular computer algebra systems like Mathematica, 27] or Macaulay, [28]. For large equation systems we found the package Grobner by Wolfgang Windsteiger and Bruno Buchberger the most useful, 29] As example we give a compact description of all Golay sequences of length 16 as derived by this technique. Lemma 2 x 2 G 16 if and only if x 0 ; x 1 ; x 2 ; x 3 ; x 4 ; ....

D. Bayer and M. Stillman, "Macaulay: A computer algebra system for algebraic geometry." available by anonymous ftp from zariski.harvard.edu.

....to be given in Section 3 is more explicit, self contained and custom tailored to integer programming. Our discussion assumes familiarity with Grobner bases as presented in e.g. AL] CT] TTN] Th] We remark that software for computing them is readily available and surprisingly efficient [BaS]. The reduced Grobner basis of A with respect to c is a finite subset G c of ker Z (A) it is the minimal test set (in the sense of [Schr] x17.3) for all programs IP A;c ( Delta) The union of all reduced Grobner bases G c , as c varies over R n , is a finite set. It is denoted UGBA and called ....

....if and only if x u does not lie in in c (I A ) In what follows, we use in c (I A ) to denote [ t i=1 (ff i N n ) An important motivation for the use of Grobner bases in integer programming is the existence of computer programs for calculating them. The general purpose package MACAULAY [BaS] worked well for many non trivial computations, such as the ones listed in the table in Example 2.12 below. More specialized Grobner basis software for integer programming is currently being developed by Serkan Hosten at Cornell University [HS] Example 1.2 (continued) For c 0 = 1; 0; 0; 1; 0; ....

D. Bayer and M. Stillman, MACAULAY: A computer algebra system for algebraic geometry. Available by anonymous ftp from zariski.harvard.edu.

....the possible values of x 1 . A final substitution into g 1 exhibits all the solutions of our original system of equations. 3 Algorithms for performing Grobner bases calculations have been implemented in many popular computer algebra systems like Mathematica, 39] Maple, 40] or Macaulay, [41]. For large equation systems we found the package Grobner by Wolfgang Windsteiger and Bruno Buchberger the most useful, 42] The following example demonstrates how Grobner bases techniques can be applied to find (and compactly describe) Golay sequences. Example 10 Assume we want to find all ....

D. Bayer and M. Stillman, "Macaulay: A computer algebra system for algebraic geometry." available by anonymous ftp from zariski.harvard.edu.

....such as Algorithm 2 due to Urbanke and DiBiase are also implemented. Both algorithms produce Grobner bases for toric ideals considerably faster than any previously known method. While efficient software to do Grobner basis computations for general ideals is readily available (e.g. MACAULAY [2]) it is our aim to exploit the specific structure of toric ideals to achieve maximum speed. First experiments in the same direction within the system ALPI were reported in [4] A more systematic study and implementation was carried out by C.Moulinet [10] who ran many experiments. When running ....

D. Bayer & M. Stillman, Macaulay: a computer algebra system for algebraic geometry, available by anonymous ftp from zariski.harvard.edu.

....encode an IP problem into a special ideal associated with the constraint matrix A and the cost (object) function C. An important property of such an encoding is that its Grobner bases correspond directly to the test sets of the IP problem. Thus, by employing an algebraic package such as MACAULAY [4] or MAPLE [6] the test sets of the IP problem can be directly computed. Using a proper test set (such as the minimal test set which corresponds directly to the reduced Grobner base of the ideal) the optimal value of the cost function can be computed by constructing a monotonic path from the ....

D. Bayer and M. Stillman, MACAULAY: A Computer Algebra System for Algebraic Geometry . Available by anonymous ftp from zariski:harvard:edu:

....(b) are both non trivial. We will briefly address the former problem in Section 7. Once these two issues are taken care of, the reduced Grobner basis G c and the normal form of x u can be computed using a software package that does Grobner basis computations. Two such packages are MACAULAY [BS] and GRIN [HS] of which the latter is custom tailored for integer programming. We note that Algorithm 2.4 is a condensed version of the original Conti Traverso algorithm in [CT] This version is useful for highlighting the main computational steps involved, although the original version is more ....

D. Bayer and M. Stillman, Macaulay: a computer algebra system for algebraic geometry, available by anonymous ftp from zariski.harvard.edu. New version Macaulay2 by D. Grayson and M. Stillman available from http://www.math.uiuc.edu/¸dan/.

....which is large for most problems. Due to their structure such methods have advantages over other more conventional IP methods in solving stochastic integer programming problems; see Schultz et al. 48] Computer packages for computing Gr obner bases are available, e.g. CoCoa [16] and MACAULAY [13]. 5 Polynomial time approximation As an alternative to solving NP hard combinatorial optimization problems to optimality, which may be very time consuming, a stream of research has concentrated on designing polynomial time algorithms that aim at good approximations for such problems. A widely ....

D. Bayer, M. Stillman. MACAULAY: A Computer Algebra System for Algebraic Geometry, Available by anonymous ftp from zariski.harvard.edu.

....Grobner basis required for the proposed algorithm has 22 elements, compared to 1180 elements that the standard algorithm requires. The listed running times could be further reduced in several ways. Instead of using Mathematica to perform the Grobner basis calculation, one could use Macaulay [Bayer and Stillman] which is significantly faster. More substantially, all occurring ideals are toric ideals, for which specialized Buchberger algorithms have been investigated [Conti and Traverso 1991; Hosten and Sturmfels 1994] These speedups apply to the standard as (n; m) standard alg. present alg. 3; 2) ....

D. Bayer and M. Stillman, Macaulay: A Computer Algebra System for Algebraic Geometry. This manual is part of the program distribution, available by anonymous ftp from the host available by anonymous ftp from zariski.harvard.edu.

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D. Bayer and M. Stillman. Macaulay: a computer algebra system for algebraic geometry. available by anonymous ftp from ftp.math.harvard.edu.

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D. Bayer and M. Stillman, Macaulay: A computer algebra system for algebraic geometry. Software, 1994. Available at www.math.columbia.edu/#bayer/Macaulay/.

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Bayer, D., and Stillman, M., Macaulay: a computer algebra system for algebraic geometry, available by anonymous ftp from zariski.harvard.edu.

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D. Grayson and M. Stillman. Macaulay 2: a computer algebra system for algebraic geometry, Version 0.8.56, www.math.uiuc.edu/macaulay2. 1999.

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D. Grayson and M. Stillman. Macaulay 2: a computer algebra system for algebraic geometry, Version 0.8.56, www.math.uiuc.edu/macaulay2. Scripts by A. Leykin and H. Tsai, www.math.berkeley.edu/~htsai. 1999.

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Bayer, D. and Stillman, M., Macaulay: A computer algebra system for algebraic geometry,

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