What is it about?

We construct an example of a smooth convex function on the plane with a strict minimum at zero, which is real analytic except at zero, for which Thom's gradient conjecture fails both at zero and infinity. More precisely, the gradient orbits of the function spiral around zero and at infinity. Besides, the function satisfies the Lojasiewicz gradient inequality at zero.

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Why is it important?

It is known that orbits of a real-analytic function satisfy the Thom gradient conjecture (i.e. the secants are asymptotically stabilized). In this work we present an example of a smooth function defined on the plane, which is real-analytic outside zero (its unique critical point), but fails the gradient conjecture both at zero and at infinity. This example outlines the two failures of o-minimality of the function, despite the fact that the function is convex and satisfies the Lojasiewicz gradient inequality.


Semialgebraic functions satisfy the Thom gradient conjecture at infinity. It is not unknown if this remains true for orbits of any polynomially bounded o-minimal function (or more generally, for any o-minimal function).

Aris Daniilidis
TU Wien

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This page is a summary of: A convex function satisfying the Łojasiewicz inequality but failing the gradient conjecture both at zero and infinity, Bulletin of the London Mathematical Society, March 2022, Wiley, DOI: 10.1112/blms.12586.
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