Quadratic polynomial maps with Jacobian rank two

Polynomial maps with strongly nilpotent Jacobian matrix and the Jacobian Conjecture

Identifiability of an X-rank decomposition of polynomial maps

In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing and machine learning. We show that this decomposition is a special case of the Xrank decomposition — a powerful novel concept in algebraic geometry that generalizes the tensor CP decomposition. We prove new results on generic/maximal rank and on identifiability of the pol...

A new approach to interior regularity of elliptic systems with quadratic Jacobian structure in dimension two

then u is continuous. This result solves a conjecture of Heinz about regularity of solutions to the prescribed boundedmean curvature equation and a conjecture of Hildebrandt about regularity of all critical points of continuously differentiable elliptic conformally invariant Lagrangians in dimension two. In particular it provides a new proof of Hélein’s theorem [10,12] about regularity of two d...

Quadratic Maps with Convex Images

Seeking exactness for \convex relaxation", we deene a map f : < n ! < m to be ICON if its image f(< n) is convex. Analogously, f is termed LICON(AICON) if the image of every linear (every aane) vsubspace under f is convex. In this paper, we investigate quadratic maps of these three types. These maps are motivated by a fact shown here that they prescribe conditions under which feasibility and op...

The Jacobian Conjecture: linear triangularization for homogeneous polynomial maps in dimension three

Let k be a field of characteristic zero and F : k → k a polynomial map of the form F = x + H, where H is homogeneous of degree d ≥ 2. We show that the Jacobian Conjecture is true for such mappings. More precisely, we show that if JH is nilpotent there exists an invertible linear map T such that T−1HT = (0, h2(x1), h3(x1, x2)), where the hi are homogeneous of degree d. As a consequence of this r...