Journal ArticleDOI
The precise determination of Maxwell sets for cuspoid catastrophes
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This paper establishes algebraic conditions for the polynomial to have a pair of minima that have a common horizontal tangent, exactly that required by the Maxwell convention of catastrophe theory.Abstract:
Given a polynomial in a single variable, we can find algebraic conditions on the coefficients corresponding to geometrically apparent features of its graph. For example, in elementary catastrophe theory, a bifurcation set in the coefficient space corresponds to a flat point of inflection on the curve. In this paper we establish algebraic conditions for the polynomial to have a pair of minima that have a common horizontal tangent. This condition is exactly that required by the Maxwell convention of catastrophe theory. The first step is a proof of a result giving a factorization of the discriminant of the discriminant of a polynomial. The proof given is a direct algebraic factorization. This result provides a condition for equal critical values. To narrow down to the cases in which the equal values are both minima, a method of using a combination of Sturm sequences has been devised. The paper thus provides complete rigorous algorithm for the determination of an algebraic expression for the Maxwell set of a ...read more
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Book ChapterDOI
Sparse Discriminants and Applications
TL;DR: This work describes discriminants in a general context, and relates them to an equally useful object, namely the resultant of an overconstrained polynomial system, and focuses on exploiting the sparseness of polynomials via the theory of Newton polytopes and sparse elimination.
Journal ArticleDOI
Determining a generalized Maxwell set
TL;DR: In this article, the authors consider the condition that a polynomial has critical values that differ by a given amount and derive algebraic expressions corresponding to this condition for the cusp catastrophe.
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Book
Theory of Equations.
Abstract: If f(x) represents the function a1x n + a2x n −1 + … + an + 1 and f(a) denotes the value of the function when x is replaced by the value a, then if
$$\matrix{ {f\left( x \right) = 2{x^3} - 3{x^2} + 5x - 6} \cr {f\left( 1 \right) = 2 - 3 + 5 - 6 = - 2} \cr {f\left( { - 1} \right) = - 2 - 3 - 5 - 6 = - 16} \cr } $$
and
$$f\left( {2.5} \right) = \ldots \ldots \ldots .$$
for
$$\matrix{ {f\left( {2.5} \right) = 2{{\left( {2.5} \right)}^3} - 3{{\left( {2.5} \right)}^2} + 5\left( {2.5} \right) - 6} \cr { = 31.25 - 18.75 + 12.5 - 6 = \underline {19} } \cr } $$