6
$\begingroup$

My colleague and I recently wanted to find a form of the Implicit Function Theorem where there were explicit lower bounds on the sizes of the neighborhood of validity (for input and output). The textbooks we have looked at so far do not seem to give such bounds, e.g.:

  • Rudin, Principles of Mathematical Analysis (3rd ed.), Thm. 9.28, pp. 224-225: "there exist open sets $U \subset R^{n + m}$ and $W \subset R^{m}$ such that ..."
  • Fritzsche-Grauer, From Holomorphic Functions to Complex Manifolds (GTM 213), Thm. 7.6, p. 34: "Then there is an open neighborhood $U = U^{\prime} \times U^{\prime \prime} \subset B$ and a holomorphic map $g: U^{\prime} \to U^{\prime \prime}$ such that ..."
  • Krantz-Parks, The Implicit Function Theorem: History, Theory, and Applications, Thm. 6.1.2, p. 118: "there exist $r_0 > 0$ and a power series ..." and the proof simply says (p. 121) "and $r$ is sufficiently small (depending on $R_1$ and $R_2$)."

A quick internet search for "Quantitative Implicit Function Theorem" also does not reveal any sources that give me quite what I want:

  • Azzam-Schul, "Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps", https://arxiv.org/abs/1105.4198 , shows that up to a small-Hausdorff-Content set, a $1$-Lipschitz map $[0, 1]^n \to \mathbb{R}^n$ partitions into Lipschitz pieces that can each be extended to $\mathbb{R}^n$ in the domain. I would like a statement that gives me a complete-ball neighborhood, not a neighborhood up to small-content pieces.
  • Phien, "Some quantitative results on Lipschitz inverse and implicit functions theorems," https://arxiv.org/abs/1204.4916 , gets pretty close, but in the relevant result (Thm. 3.5, I believe), one has to manually add in the hypothesis that the (Generalized) Jacobian at all points near $x$ is within a particular neighborhood of the Jacobian at $x_0$, i.e., I lose the quantitative estimate on the radius $r$ of validity.

(Phien mentions as a source Henrici, Applied and Computational Complex Analysis, vol. 1, but the library copies are checked out.)

The only reference I have that is close to what I want is: Y. Yomdin, "Some Quantitative Results In Singularity Theory", Annales Polonici Mathematici 87(2005), 277-299 https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/annales-polonici-mathematici/all/87//84792/some-quantitative-results-in-singularity-theory Theorem 2.2. and its proof (pp. 7 - 8) state that for $f(x, y)$ a real-analytic function with $f(0, 0) = 0$ and a power series $f(x, y) = \sum_{k, \ell = 1}^{\infty} a_{k, \ell} x^k y^{\ell}$ with $a_{0, 1} \neq 0$, one infers the existence of the function $y = h(x)$ on the domain $|x| \leq R_1$, with image satisfying $|h(x)| \leq R^{\prime}$, where $R_1$, $R^{\prime}$ are defined explicitly in the proof. Indeed, we can use it in our case. I am still slightly dissatisfied, however, as the given constants depend on the growth rates of the coefficients, and in particular, require more than the first 1-2 partial derivatives one expects to use in Implicit Function Theorems.


The Question

Is there a reputable reference to something like the following:

Let $\mathbf{T}$, $\mathbf{Z}$ be nice enough (closures of?) open domains in $\mathbb{C}$. Let $F(t, z): \mathbf{T} \times \mathbf{Z} \to \mathbb{C}$ be a complex-analytic (or $C^3$, or ...) function. Moreover, suppose that $F(t_0, z_0) = 0$ and we have both that $$ \left\vert \frac{\partial F}{\partial z} \right\vert \geq c > 0,$$ at least near the zero-set $\lbrace (t, z): F(t, z) = 0 \rbrace$, and that the first and second (possibly the third) partial derivatives are bounded above in an appropriate neighborhood. Then there exist positive $r$ and $\kappa$, depending only on $c$ and the upper bounds on the magnitudes of the derivatives, and a function $h: B_{r}(t_0) \to \mathbb{C}$ such that

  1. $h(t_0) = z_0$
  2. $F(t, h(t)) = 0, \, \, t \in B_r(t_0)$
  3. $|h(t)| \leq \kappa, \, \, t \in B_r(t_0)$.

In short, just the standard Implicit Function Theorem, but with minimum sizes for the neighborhoods of validity, depending on the derivative bounds.

Of course, more general statements for higher dimensions would be nice, but I would really like to be able to use complex Jacobians directly.

$\endgroup$
  • 1
    $\begingroup$ This would be nice to find out.... If memory serves the book by Hubbard and Hubbard might have something on this. $\endgroup$ – Chris Jun 7 '17 at 2:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.