# On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

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?

• 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
• 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 0:15
• @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) – MathOverview Jul 11 '13 at 13:02
• @Matt I modified my question. I clarified what I mean by counterexample. Thank you. – MathOverview Jul 11 '13 at 13:07
• 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