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Characteristic values of the Jacobian matrix and global invertibility

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Campbell L.

Annales Polonici Mathematici

2001

tom 76

nr 1-2

11-20

Characteristic matrix values (singular values, eigenvalues, and pivots arising from Gaussian elimination) for the Jacobian matrix and its inverse are considered for maps of real n-space to itself with a nowhere vanishing Jacobian determinant. Bounds on these are related to global invertibility of the map. Polynomial maps with a constant nonzero Jacobian determinant are a special case that allows for sharper characterizations.

Reduction theorems for the Strong Real Jacobian Conjecture

100%

Campbell L.

Annales Polonici Mathematici

2014

tom 110

nr 1

1-11

Implementations of known reductions of the Strong Real Jacobian Conjecture (SRJC), to the case of an identity map plus cubic homogeneous or cubic linear terms, and to the case of gradient maps, are shown to preserve significant algebraic and geometric properties of the maps involved. That permits the separate formulation and reduction, though not so far the solution, of the SRJC for classes of nonsingular polynomial endomorphisms of real n-space that exclude the Pinchuk counterexamples to the SRJC, for instance those that induce rational function field extensions of a given fixed odd degree.

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2 Annales Polonici Mathematici

2 Campbell L.

1 2014

1 2001

Wyniki wyszukiwania

Characteristic values of the Jacobian matrix and global invertibility

Reduction theorems for the Strong Real Jacobian Conjecture