Stable border bases for ideals of points

Abbott, John; Fassino, Claudia; Torrente, Maria-Laura

doi:10.1016/j.jsc.2008.05.002

Cited by 32 publications

(45 citation statements)

References 9 publications

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“…which we hope you find, as we do, significantly more comprehensible as an expression describing a function from C\0 to C. Yet, for most purposes, the formulas in (1) and in (2) are equivalent: For example, the singularities are the same, and because the zeros of these expressions are wellconditioned, the locations of the corresponding zeros of the two expressions differ by less than one part in 10 5 .…”

Section: Introductionmentioning

confidence: 80%

Rounding coefficients and artificially underflowing terms in non-numeric expressions

Corless

Postma

Stoutemyer

2011

ACM Commun. Comput. Algebra

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This article takes an analytical viewpoint to address the following questions: 1. How can we justifiably beautify an input or result sum of non-numeric terms that has some approximate coefficients by deleting some terms and/or rounding some coefficients to simpler floating-point or rational numbers? 2. When we add two expressions, how can we justifiably delete more non-zero result terms and/or round some result coefficients to even simpler floating-point, rational or irrational numbers?The methods considered in this paper provide a justifiable scale-invariant way to attack these problems for subexpressions that are multivariate sums of monomials with real exponents.

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Section: Introductionmentioning

confidence: 80%

Rounding coefficients and artificially underflowing terms in non-numeric expressions

Corless

Postma

Stoutemyer

2011

ACM Commun. Comput. Algebra

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“…Proposition 6. Let f = f (x) be a degree ≥ 2 polynomial of P , let p be a point of R n such that Jac f (p) is not the zero vector, and let B ε (p) be the unit ball centered at p. Let R be a positive real number such that R < min ε min , Jac f (p) 2 H and let…”

Section: Sufficient Crossing Conditionsmentioning

confidence: 99%

“…An interesting class of recently developed algorithms relies on tools from Numerical Commutative Algebra [17,2,10,6,7]. For all these algorithms the input is a set of points possibly in n-dimensions and the output is a polynomial f in n-variables whose zero locus (which is a curve, or a surface, or more generally an algebraic variety) gives an approximation for the input points and can be interpreted as an implicit polynomial regression model [12,Ch 2].…”

Section: Step I: Approximation Of a Path By A Polynomial Curvementioning

confidence: 99%

An Euclidean norm based criterion to assess robots’ 2D path-following performance

Saggini

Torrente

2016

J Alg Stat

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Abstract. A current need in the robotics field is the definition of methodologies for quantitatively evaluating the results of experiments. This paper contributes to this by defining a new criterion for assessing path-following tasks in the planar case, that is, evaluating the performance of robots that are required to follow a desired reference path. Such criterion comes from the study of the local differential geometry of the problem. New conditions for deciding whether or not the zero locus of a given polynomial intersects the neighbourhood of a point are defined. Based on this, new algorithms are presented and tested on both simulated data and experiments conducted at sea employing an Unmanned Surface Vehicle.2000 Mathematics Subject Classifications: 26C10, 15A60, 93C85, 62P99.

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“…DRL order. 1 The difference with -Buchberger or -MatrixF5 is that it could not stop immediately when the problem happened. We counted the number of systems that have at least one polynomial g ∈ G with small leading coefficient.…”

Section: Dense Casesmentioning

confidence: 99%

Pivoting in Extended Rings for Computing Approximate Gröbner Bases

Faugère

Liang

2011

Math.Comput.Sci.

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It is well known that in the computation of Gröbner bases arbitrarily small perturbations in the coefficients of polynomials may lead to a completely different staircase, even if the solutions of the polynomial system change continuously. This phenomenon is called artificial discontinuity in Kondratyev's Ph.D. thesis. We show how such phenomenon may be detected and even "repaired" by using a new variable to rename the leading term each time we detect a "problem". We call such strategy the TSV (Term Substitutions with Variables) strategy. For a zero-dimensional polynomial ideal, any monomial basis (containing 1) of the quotient ring can be found with the TSV strategy. Hence we can use TSV strategy to relax term order while keeping the framework of Gröbner basis method so that we can use existing efficient algorithms (for instance the F 5 algorithm) to compute an approximate Gröbner basis. Our main algorithms, named TSVn and TSVh, can be used to repair artificial -discontinuities. Experiments show that these algorithms are effective for some nontrivial problems.

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Stable border bases for ideals of points

Cited by 32 publications

References 9 publications

Rounding coefficients and artificially underflowing terms in non-numeric expressions

Rounding coefficients and artificially underflowing terms in non-numeric expressions

An Euclidean norm based criterion to assess robots’ 2D path-following performance

Pivoting in Extended Rings for Computing Approximate Gröbner Bases

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