On the total order of reducibility of a pencil of algebraic plane curves

In this paper, the problem of bounding the number of reducible curves in a pencil of algebraic plane curves is addressed. Unlike most of the previous related works, each reducible curve of the pencil is here counted with its appropriate multiplicity. It is proved that this number of reducible curves, counted with multiplicity, is bounded by d^2-1 where d is the degree of the pencil. Then, a sharper bound is given by taking into account the Newton's polygon of the pencil

On the Regularity Property of Differential Polynomials Modulo Regular Differential Chains

International audienceThis paper provides an algorithm which computes the normal form of a rational differential fraction modulo a regular differential chain if, and only if, this normal form exists. A regularity test for polynomials modulo regular chains is revisited in the nondifferential setting and lifted to differential algebra. A new characterization of regular chains is provided

Need Polynomial Systems Be Doubly-Exponential?

Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables [Mayr and Mayer, 1982], [Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that number of zeros of a zero-dimensional system is singly-exponential in the number of variables. How should this contradiction be reconciled? We first note that [Mayr and Ritscher, 2013] shows that the doubly exponential nature of Gr\"{o}bner bases is with respect to the dimension of the ideal, not the number of variables. This inspires us to consider what can be done for Cylindrical Algebraic Decomposition which produces a doubly-exponential number of polynomials of doubly-exponential degree. We review work from ISSAC 2015 which showed the number of polynomials could be restricted to doubly-exponential in the (complex) dimension using McCallum's theory of reduced projection in the presence of equational constraints. We then discuss preliminary results showing the same for the degree of those polynomials. The results are under primitivity assumptions whose importance we illustrate.Comment: Extended Abstract for ICMS 2016 Presentation. arXiv admin note: text overlap with arXiv:1605.0249

Blunted angiogenesis and hypertrophy are associated with increased fatigue resistance and unchanged aerobic capacity in old overloaded mouse muscle.

We hypothesize that the attenuated hypertrophic response in old mouse muscle is (1) partly due to a reduced capillarization and angiogenesis, which is (2) accompanied by a reduced oxidative capacity and fatigue resistance in old control and overloaded muscles, that (3) can be rescued by the antioxidant resveratrol. To investigate this, the hypertrophic response, capillarization, oxidative capacity, and fatigue resistance of m. plantaris were compared in 9- and 25-month-old non-treated and 25-month-old resveratrol-treated mice. Overload increased the local capillary-to-fiber ratio less in old (15 %) than in adult (59 %) muscle (P < 0.05). Although muscles of old mice had a higher succinate dehydrogenase (SDH) activity (P < 0.05) and a slower fiber type profile (P < 0.05), the isometric fatigue resistance was similar in 9- and 25-month-old mice. In both age groups, the fatigue resistance was increased to the same extent after overload (P < 0.01), without a significant change in SDH activity, but an increased capillary density (P < 0.05). Attenuated angiogenesis during overload may contribute to the attenuated hypertrophic response in old age. Neither was rescued by resveratrol supplementation. Changes in fatigue resistance with overload and aging were dissociated from changes in SDH activity, but paralleled those in capillarization. This suggests that capillarization plays a more important role in fatigue resistance than oxidative capacity

Resultant of an equivariant polynomial system with respect to the symmetric group

Given a system of n≥ 2 homogeneous polynomials in n variables which is equivariant with respect to the symmetric group of n symbols, it is proved that its resultant can be decomposed into a product of several resultants that are given in terms of some divided differences. As an application, we obtain a decomposition formula for the discriminant of a multivariate homogeneous symmetric polynomial. © 2015 Elsevier Ltd

Implicitizing rational curves by the method of moving quadrics

A new technique for finding implicit matrix-based representations of rational curves in arbitrary dimension is introduced. It relies on the use of moving quadrics following curve parameterizations, providing a high-order extension of the implicit matrix representations built from their linear counterparts, the moving planes. The matrices we obtain offer new, more compact, implicit representations of rational curves. Their entries are filled by linear and quadratic forms in the space variables and their ranks drop exactly on the curve. Typically, for a general rational curve of degree d we obtain a matrix whose size is half of the size of the corresponding matrix obtained with the moving planes method. We illustrate the advantages of these new matrices with some examples, including the computation of the singularities of a rational curve. © 2019 Elsevier Lt

Resultant Over the Residual of a Complete Intersection

In this article, we study the residual resultant which is the necessary and sufficient condition for a polynomial system F to have a solution in the residual of a variety, defined here by a complete intersection G. We show that it corresponds to an irreducible divisor and give an explicit formula for its degree in the coefficients of each polynomial. Using the resolution of the ideal (F : G) and computing its regularity, we give a method for computing the residual resultant using a matrix which involves a Macaulay and a Bezout part. In particular, we show that this resultant is the gcd of all the maximal minors of this matrix. We illustrate our approach for the residual of points and end by some explicit examples