Peano on Wronskians: A Translation

Engdahl, Susannah M.; Parker, Adam E.

doi:10.4169/loci003642

Cited by 2 publications

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“…But the converse is not necessarily true. Peano, then Bôcher, found counterexamples [21,22,5] (see [8] for a history of these results). However, Bôcher [6] proved that this converse becomes true if the functions are analytic [15].…”

Section: Zeros Of the Wronskian As An Upper Boundmentioning

confidence: 99%

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A Wronskian approach to the real τ-conjecture

Koiran

Portier

Tavenas

2015

Journal of Symbolic Computation

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According to the real τ -conjecture, the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower bound on the arithmetic circuit complexity of the permanent.In this paper, we use the Wronksian determinant to give an upper bound on the number of real roots of sums of products of sparse polynomials of a special form. We focus on the case where the number of distinct sparse polynomials is small, but each polynomial may be repeated several times. We also give a deterministic polynomial identity testing algorithm for the same class of polynomials.Our proof techniques are quite versatile; they can in particular be applied to some sparse geometric problems that do not originate from arithmetic circuit complexity. The paper should therefore be of interest to researchers from these two communities (complexity theory and sparse polynomial systems).

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Section: Zeros Of the Wronskian As An Upper Boundmentioning

confidence: 99%

“…Hence P (Y ) has a finite number of roots. By (8), Im(f) is included in the (finite) set of roots of P . Nevertheless, as f is continuous, by the intermediate value theorem Im(f) is a real interval.…”

Section: An Improved Upper Boundmentioning

confidence: 99%

A Wronskian approach to the real τ-conjecture

Koiran

Portier

Tavenas

2015

Journal of Symbolic Computation

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A Wronskian Approach to Observability and the Unobservable Subspace through the Null-Space Analysis

Hagiwara

Fujii

2022

Transactions of ISCIE

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This paper discusses a connection between the Wronski matrix and observability of linear timeinvariant systems as well as their unobservable subspaces. For a linear time-invariant single-output system, its observability is closely related to linear independence of a set of functions associated with the output. Hence, there is a connection between observability and the Wronski matrix, which is often used for deciding linear independence of a set of functions. In discussing observability and the unobservable subspace in this context, it matters whether the Wronski matrix can describe the necessary and sufficient condition for an arbitrary set of functions to be linearly independent. This paper first reviews the known facts about the relationship between linear independence of a set of functions and the Wronski matrix, as well as an additional condition under which linear independence can be decided in a necessary and sufficient fashion through the Wronski matrix. Then, well-known conditions on observability and characterization of the unobservable subspace of linear time-invariant systems are revisited from the viewpoint of the null space of the Wronski matrix.

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Peano on Wronskians: A Translation

Cited by 2 publications

References 0 publications

A Wronskian approach to the real τ-conjecture

A Wronskian approach to the real τ-conjecture

A Wronskian Approach to Observability and the Unobservable Subspace through the Null-Space Analysis

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