01-09-2023 | Preface
Further perspectives on elimination
Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 5/2023
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Excerpt
Abhyankar’s ’eliminate’ rhythmic refrain scans the climax periods in the evolution of the approach to solving by the Computational Algebraic Geometry communities, those around conferences as MEGA, ISSAC, CoCoA, JAA and which mainly publish in journals as Commutative Algebra, JSC, AAC and also AAECC:In order to save the junior reader the trouble of referring to other works, I have put together here those elementary principles with regard to elimination, of which use has been made in the preceding pages.G. Salmon A treatise on the higher plane curves Dublin 1852 pg.285The device that follows, which, it may be hoped, finally eliminates from algebraic geometry the last traces of elimination-theoryA. Weil Foundations of Algebraic Geometry A.M.S. 1962 pg.31Eliminate, eliminate, eliminateEliminate the eliminators of elimination theory.S. S. Abhyankar Polynomials and Power Series (A Poem) 1970
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In 1750 Cramer [14] gave [54](1)a rule for forming the terms of the common denominator of the fractions which express the values of the unknowns in a set of linear equations;(2)a rule for determining the sign of any individual term in the said common denominator (and, included in the rule, the notion of a “dérangement”).(3)A rule for obtaining the numerators from the expression for the common denominator
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Abhyankar’s call to arms for eliminating the eliminators of elimination theory coincides with a period when the availability of new computational tools reoriented the Computational Algebraic Geometry communities, which included also Abhyankar himself [1], to reconsider and reformulate the results of the community inspired by Hilbert introducing Buchberger’s Gröbner bases [4‐7] (see also [22]), Faugère’s \(F_4\) [20] and \(F_5\) [21] (see also [18, 19]) and Gerdt’s [23, 24] and Seiler’s [57‐59] involutive bases.