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Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the Implicit Function Theorem (finite-dimensional vector spaces) and the Picard Theorem for ordinary differential equations. We know that the Implicit Function Theorem (finite-dimensional vector spaces) is a particular case of the Constant Rank Theorem. We also know that the Frobenius Theorem is a generalization of Picard's Theorem for ordinary differential equations. Based on these facts follow my question.

The Constant Rank Theorem and the Frobenius Theorem for differential equations ( ODE's or/and PDE's) are equivalent?

Is there any reference which provides a solution to this question? If the Frobenius theorem does not imply the Constant Rank Theorem there is some explanation for the negative? Conversely, if the Constant Rank Theorem does not imply the Frobenius theorem there is some explanation for the negative?

Elias Costa
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    This is an interesting question. While on the one hand it makes sense to me how one would attempt to prove they are equivalent (show they each follow as corollaries to each other), it is incredibly confusing to think about what a counterexample would look like since these two theorems are about very different objects. It seems you could produce counterexamples to particular ideas/methods of attempted proof, but do you have any feel for what a counterexample to the statement that they are equivalent would look like at that level of generality? – Matt Jun 22 '13 at 23:56
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    Offhand, I might envision Frobenius $\implies$ Rank Theorem, but the converse is totally implausible: Given an integrable differential system, from where would a function materialize? – Ted Shifrin Jun 23 '13 at 00:15
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    @TedShifrin I believe that the function is materialized in a manner analogous to the proof of the equivalence of Picard's theorem for ODE's and the implicit function theorem in book The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) – Elias Costa Jul 11 '13 at 13:02
  • @Matt I modified my question. I clarified what I mean by counterexample. Thank you. – Elias Costa Jul 11 '13 at 13:07
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    As @Matt said, both Constant Rank Theorem and Fobrenius Theorem for differential equations open in different directions. To compare them it would be good if you put both statements in a common context – Daniel Camarena Perez Dec 10 '18 at 15:10
  • It would help the question if you wrote an explanation of the equivalence between Picard's theorem for ODEs and the implicit function theorem, in particular how the function for the implicit function theorem is materialized and why you think it can be extended to this more general case, rather than just a reference to the book and noting that both the constant rank theorem and the frobenius theorem are generalizations of two things that were shown to be equivalent. – Jürgen Sukumaran Mar 17 '23 at 09:45
  • Maybe an idea to start a solution is start from some differential equation where the Picard theorem fail and later see if are still fulfilled the other theorems: like example review what happen with the Norton's dome example's differential equation where uniqueness is broken, or maybe something easier like: $$x' = -\text{sgn}(x)\sqrt{|x|}$$ which allows unique solutions $$x(t) = \frac{\text{sgn}(x(0))}{4}\left(2\sqrt{|x(0)|}-t\right)^2\cdot\theta!\left(2\sqrt{|x(0)|}-t\right)$$ but the differential equation has similar issues than Norton's one. – Joako Sep 13 '23 at 21:51
  • in my previous comment $\theta(t)$ is the unitary step function (click here to see the example fulfill uniqueness). – Joako Sep 13 '23 at 21:53
  • In the provided book "The Implicit Function Theorem: History, Theory, and Applications" by Steven G. Krantz and Harold R. Parks, it seems that the content you were reading was truncated, and page 52 is not available. Therefore, it is challenging to provide specific information from that page. – Furdzik Zbignew Sep 22 '23 at 12:34

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