https://wiki.sarg.dev/index.php?action=history&feed=atom&title=Gradient_conjecture 2026-04-24T23:08:08Z Revision history for this page on the wiki MediaWiki 1.44.2 https://wiki.sarg.dev/index.php?title=Gradient_conjecture&diff=363094&oldid=prev 2025-04-20T03:08:05Z

<p>Moving <a href="/index.php?title=Category:Theorems_in_analysis&amp;action=edit&amp;redlink=1" class="new" title="Category:Theorems in analysis (page does not exist)">Category:Theorems in analysis</a> to <a href="/index.php?title=Category:Theorems_in_mathematical_analysis&amp;action=edit&amp;redlink=1" class="new" title="Category:Theorems in mathematical analysis (page does not exist)">Category:Theorems in mathematical analysis</a> per <a href="https://en.wikipedia.org/wiki/Categories_for_discussion/Log/2025_April_12#Category:Theorems_in_analysis" class="extiw" title="wikipedia:Categories for discussion/Log/2025 April 12">Wikipedia:Categories for discussion/Log/2025 April 12#Category:Theorems in analysis</a></p> <p><b>New page</b></p><div>In [[mathematics]], the &#039;&#039;&#039;gradient conjecture&#039;&#039;&#039;, due to [[René Thom]] (1989), was proved in 2000 by three Polish mathematicians, Krzysztof Kurdyka ([[University of Savoie]], France), Tadeusz Mostowski ([[Warsaw University]], Poland) and Adam Parusiński ([[University of Angers]], France).<br /> <br /> The conjecture states that given a real-valued [[analytic function]] &#039;&#039;f&#039;&#039; defined on R&lt;sup&gt;&#039;&#039;n&#039;&#039;&lt;/sup&gt; and a [[trajectory]] &#039;&#039;x&#039;&#039;(&#039;&#039;t&#039;&#039;) of the [[gradient]] vector field of &#039;&#039;f&#039;&#039; having a [[limit point]] &#039;&#039;x&#039;&#039;&lt;sub&gt;0&lt;/sub&gt; ∈ R&lt;sup&gt;&#039;&#039;n&#039;&#039;&lt;/sup&gt;, where &#039;&#039;f&#039;&#039; has an isolated critical point at &#039;&#039;x&#039;&#039;&lt;sub&gt;0&lt;/sub&gt;, there exists a limit (in the [[projective space]] PR&lt;sup&gt;&#039;&#039;n&#039;&#039;−1&lt;/sup&gt;) for the [[secant line]]s from &#039;&#039;x&#039;&#039;(&#039;&#039;t&#039;&#039;) to &#039;&#039;x&#039;&#039;&lt;sub&gt;0&lt;/sub&gt;, as &#039;&#039;t&#039;&#039; tends to zero.<br /> <br /> The proof depends on a theorem due to [[Stanis%C5%82aw %C5%81ojasiewicz]].<br /> <br /> ==References==<br /> <br /> * R. Thom (1989) &quot;Problèmes rencontrés dans mon parcours mathématique: un bilan&quot;, [[Publications Math%C3%A9matiques de l%27IH%C3%89S]] 70: 200 to 214. (This gradient conjecture due to René Thom was in fact well known among specialists by the early 70&#039;s, having been often discussed during that period by Thom during his weekly seminar on singularities at the [[IHES]].)<br /> * In 2000 the conjecture was proven correct in [[Annals of Mathematics]] 152: 763 to 792. The proof is available [https://arxiv.org/abs/math.AG/9906212 here].<br /> <br /> [[Category:Theorems in mathematical analysis]]<br /> <br /> {{Mathanalysis-stub}}</div>

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