# Questions tagged [implicit-function-theorem]

The implicit-function-theorem tag has no usage guidance.

48
questions

2
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### Function composition inside implicit function equation

For a function $f: R^n \times R^n \times R^n \rightarrow R^n$ analytic in a neighborhood of $(0,0,0)$ such that $f(0,0,0)=0$, I would like to show the existence of a function $\varphi$ analytic on a ...

3
votes

0
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111
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### Analytic analogue of implicit functions for differential operators

Let $p\colon \mathbb{R}^2 \to \mathbb{R}$ be a polynomial with a non-vanishing gradient at $p^{-1}(0)$. Then, the implicit function theorem says that $S = \{(x,y) \in \mathbb{R}^2 \mid p(x,y) = 0\}$ ...

1
vote

1
answer

81
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### Second order differentiability of solution operator to nonlinear boundary value problem

I am currently reading the paper [1]. The paper deals with the BVP:
\begin{align*}
\partial_t u - \Delta u + u(u-a) & = -qu \text{ in } ]0,T\mathclose[ \times \Omega \\[6pt]
\partial_{n}u & = ...

0
votes

0
answers

66
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### Implicit function theorem for non $C^1$ mappings

I know that the inverse function theorem can be proved for differentiable mappings (not $C^1$) by requiring that $Df(x)$ has everywhere maximum rank (here is the reference https://terrytao.wordpress....

4
votes

2
answers

302
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### Implicit function theorem without uniqueness?

Imagine you are given $f(x,y) := y^2-\sin(x)^2$
and you want to answer the question, if there is a neighbourhood of $x=0$ such that $f(x,y(x))=0$ with $y(0)=0$.
One idea that comes to mind is the ...

0
votes

1
answer

95
views

### Recovering the openness of a map from the openness of its scalar projections

Good morning. I have been thinking about the following question for a while without much success, therefore I'm starting to doubt its validity, although I don't have a clear counterexample in mind.
...

1
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0
answers

116
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### Implicit function theorem / Implicit selections when Jacobian not invertible

I saw the attached result in the book by Dontchev and Rockafellar.
It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns ...

1
vote

1
answer

94
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### Maximizing the ratio of multilinear polynomials

Consider two multilinear Polynomials $A(x_1,x_2,x_3,\dotsc,x_n)$ and $B(x_1,x_2,x_3,\dotsc,x_n)$ of $n > 2$ variables $x_i \in \mathbb{R}$ and their ratio
\begin{equation*}
F(x_1,x_2,x_3,\dotsc,...

1
vote

0
answers

258
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### Implicit function theorem and compactification of algebraic curve

Let $C$ be a singular curve defined over a local field $K$.
Let $\tilde{C}$ be its smooth compactification(maybe this is not normalization).
Why $\tilde{C}(K)\neq \emptyset$ implies ${C}(K)\neq \...

5
votes

1
answer

301
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### Implicit function theorem with singularities of any order

Let $\mathcal{U}\subset \mathbb{R}\times \mathbb{C}$ a neighborhood of $(0,0)$, and $f:\mathcal{U}\to \mathbb{C}$ differentiable in the first variable and holomorphic in the second variable, with $f(0,...

0
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0
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152
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### Implicit function theorem on curves

I am trying to figure out, whether the IFT can be generalized to curves. Let's say I have a function $G(x,u)$ mapping $\mathbb{R}^{n+m}\rightarrow \mathbb{R}^n$ with invertible jacobian $\frac{\...

0
votes

0
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386
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### Implicit function theorem for polynomials

I am looking at this version of the implicit function theorem (taken from Krantz, "Function theory of several complex variables").
If I assume that every $f_j$ is algebraic function (or ...

2
votes

2
answers

261
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### Full-rank matrix

I have a sparse square matrix and want to see if it is full rank (so that I can apply the implicit function theorem).
$$\left[\begin{array}{cccccccccc}
0 & 1 & 1 & 1 & 0 & 0 & ...

10
votes

2
answers

712
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### Implicit function theorem with continuous dependence on parameter

Let $X,Y$ be Hilbert spaces and $P$ a topological space$^1$ and $p_0\in P$.
Let $f:X\times P\to Y$ be a continuous map such that
for any parameter $p\in P$, $f_p:= f|_{X\times \{p\}}:X\to Y$ is ...

1
vote

1
answer

323
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### Does the Implicit Function Theorem in Banach spaces holds if the differential is only one-to-one (not onto!)?

Is the Implicit Function Theorem in the following form correct:
Let $V_1,V_2,W$ be Banach spaces, and $Ω⊂V_1×V_2$ an open subset containing $(x_0,y_0)$. Let consider a continuously differentiable map $...

0
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0
answers

107
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### Implicit function theorem when $dF/dy = 0$ but under monotonicity constraint of the implicit function $y(x)$

I am looking for an extended version of the implicit/inverse function theorem that would show uniqueness of a strictly increasing implicit function, even when the derivative condition is violated (e.g....

2
votes

1
answer

203
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### Implicit function theorem for stochastic differential equation

For each $\theta\in \mathbb{R}$,
we consider a stochastic differential equation (SDE):
$$
d X_t =b(t,X_t,\theta)dt+\sigma dW_t,\; t\in [0,T];\quad X_0=x_0\in \mathbb{R},
$$
where $\sigma\ge 0$ and ...

0
votes

0
answers

58
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### Arranging the $k$ solutions of $r(z)=te^{i\theta}$ into $k$ continuous functions of $(t,\theta)$

I have originally opened this question on MSE, but I migrated here, since I realized this environment is more suitable.
Let $r$ be a rational function, that is, quotient of two coprime polynomials $p,...

1
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0
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64
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### Is there a version of Hamilton's infinite dimensional family of implicit function theorems which gives us a submersion map?

Hamilton, in his notes on "Inverse Function Theorem of Nash and Moser" states a theorem(1.1.3 on Page 172), where a given nonlinear map between tame Frechet spaces is locally surjective, if ...

2
votes

1
answer

246
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### Question about the implicit function theorem. an example of a homogeneous form for which its implicit function satisfies certain conditions

Let $F$ be a homogeneous form with coefficients in $\mathbb{R}$. Suppose it defines a smooth projective variety, in other words at every point other than the origin at least one of the first partial ...

3
votes

1
answer

168
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### Tangent space of smooth Hilbert submanifolds

Let $X, Y$ be Hilbert spaces and $F:X \rightarrow Y$ smooth. Assume that $M := F^{-1}(0) \subset X$ is a smooth submanifold. Is it true that for any $x\in M$, the tangent space $T_xM$ is a Hilbert ...

4
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2
answers

264
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### Implicit function theorem for subdifferentiable convex functions

I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function ...

2
votes

0
answers

239
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### Continuity of a constrained parameterized convex optimization problem

Consider the parameterized optimization problem:
\begin{align}
\boldsymbol{s}(p)= &\arg \min_{ \boldsymbol{x}} \quad g( \boldsymbol{x})\\
\text{s.t. } & \boldsymbol{A}(p) \textbf{x} = \...

1
vote

1
answer

372
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### Is this operator invertible?

Let $T(t)$ be a strongly continuous semi-group on a Banach space $X$, and let $A(\cdot)\in C(0,\tau; \mathcal{L}(X))$ for some $\tau>0$. The operator $G:C(0,\tau;X)\to C(0,\tau;X)$ maps every $h\in ...

3
votes

1
answer

756
views

### Continuity of a parameterized convex optimization problem

I have a parameterised optimization problem:
\begin{align}
\boldsymbol{S}(p)= &\arg \min_{ \boldsymbol{x}} g( \boldsymbol{x})\\
\text{s.t. } & \boldsymbol{A}(p) \textbf{x} = \...

12
votes

0
answers

653
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### Understanding a certain algebraic set arising in Deep Learning

I'm not a professional geometer. Thanks in advance for your patience.
So, let $n$, $k$, $p_0,\ldots,p_{k}$ be positive integers. Let $X$ (resp. $Y$) be an $p_0$-by-$n$ (resp. an $p_{k}$-by-$n$) real ...

1
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0
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238
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### An application of Implicit Function Theorem for Curve Shortening Flow on plane

I'm trying understand the article "Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem" by Ben Andrews and Paul Bryan and they stated that
...

1
vote

1
answer

324
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### Continuity of implicitly defined function

Consider a function $g(x)$ defined implicitly via
$$\int_x^{x + g(x)} f(\xi) \,d \xi - u(x) = 0. $$
I know that for every $x$ a unique $g(x)$ exists.
Furthermore $f$ is locally integrable and $u$ is ...

3
votes

1
answer

129
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### Continuity in the roots of an algebraic variety with respect to the coordinates

If I'm slightly misusing definitions forgive me I'm not an algebraist.
I have $N$ polynomials $f_n(x)$, $n=1,\ldots,N$ where $x\in \mathbb{R}^N$ and the set $\{x:f_n(x)=0\text{ for all }n\}$ is ...

2
votes

0
answers

243
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### Implicit Function Theorem, parametrized - how can we get uniform domains? (from math.se)

(This question is a duplicate from here)
Consider a family of continously differentiable functions $F_r\colon\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (where $r\in[0,1]$). For every parameter $r$, we ...

0
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0
answers

262
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### Implicit Function Theorem

Let $f$ be a $C^2$ function defined on a neighborhood of $0$ in $\mathbb R^n$ such that $f(0)=0, df(0)\not=0$. By the Implicit Function Theorem, it is easy to get (after a rotation) that near 0
$$
f(x)...

2
votes

0
answers

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### Roots of a partially holomorphic function

Let $\Omega$ be an open subset of $\mathbb R^d$, let $U$ be an open subset of $\mathbb C$ and let $f:\Omega\times U\rightarrow\mathbb C$ be a $C^\infty$ function which is holomorphic with respect to $\...

2
votes

0
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356
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### Inverse and implicit function theorems with domain

I have seen an author use the Implicit Function Theorem for a map whose second partial derivative has a bounded inverse, but is unbounded. The map itself is not defined on an open set, but only on a ...

2
votes

3
answers

375
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### Compute inverse series for implicit equation $b=-\log(1-e^{-x})/x$

In financial mathematics, the inverse series of: $$b(x) = -\frac{\log(1-e^{-x})}{x}$$ is needed in order to perform fast calculation on swaptions for G2++ calibration model. (see this post for ...

1
vote

1
answer

663
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### Taylor Series expansion for an implicitely defined family of functions

Can we find a Taylor Series expansion for $y(x)$ implicitly defined by:
$$\sum _{i=1}^nA_ie^{a_ix+b_iy} = 1 ?$$
In financial mathematics, the two-additive-factors Model G2++ is commonly used for ...

3
votes

2
answers

134
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### Is there a full-rank map with connected graph and simply connected image that is not injective?

I want to find a continuously differentiable function $F:X\to Y$, where $X\subseteq\mathbb{R}^n$, $Y\subseteq\mathbb{R}^m$ are open ($n\le m$) with
${\rm rk}\, \frac{\partial F}{\partial x}(x) = n$...

2
votes

0
answers

84
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### Implicit function theorem and selfadjoint operators

Is there a version of the implicit function theorem in some space of functions from $[0,1]$ to a Hilbert space $H$ that contains as a special case the unique solvability of the initial-value problem $\...

3
votes

0
answers

112
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### A measurable implicit function with differentiated arguments

I have encountered the need for an unusual implicit function theorem, about which I know very little. I would appreciate it if someone could help me with a few pointers.
The setup is as follows. Let $...

14
votes

2
answers

980
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### Factoring constant rank maps into a submersion and an immersion

Let $X$ and $Z$ be smooth manifolds and $\phi: X \to Z$ a smooth map so that the differential $D \phi$ is everywhere of rank $d$. Is there necessarily a $d$-fold $Y$ so that $\phi$ factors as a ...

3
votes

0
answers

322
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### Implicit function theorem for singularities

I am looking for an implicit function theorem which holds also on singular spaces, at least if the singularities are "mild".
For example, let $0 = z^2 - x y + z w + w^2 + \epsilon w$ define a ...

3
votes

1
answer

2k
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### Does the implicit function theorem hold for discontinuously differentiable functions?

(This was posted on math.SE over 5 days ago and has not been answered,
although a comment mentioned a similar question on this site.)
Wikipedia's statement of the implicit function theorem requires ...

3
votes

1
answer

579
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### Concavity of the solution of a parametric implicit function

Suppose $F(x,y;k)=f(x,y)+kg(x,y)=0$ uniquely defines the solution $y(x;k)$ for $x\in \mathbb{D}$, a compact domain, and $0\leq k \leq 1$ is a parameter. We know that for $k=0,1$, $y(x;0)$ and $y(x,1)$ ...

1
vote

1
answer

623
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### a closed form lower bound solution for linear programming

Given a linear objective function and a system of linear constraints, is there any known closed form lower bounds for it?
to clearly express the problem assume that
$$
z(\mathbf{a,B,c})=\mathop {\inf} ...

4
votes

0
answers

860
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### A strong form of implicit function theorem (what happens when the derivative is degenerate?)

(this can be considered as some ad)
Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...

1
vote

1
answer

336
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### Implicit function theorem for boundary value problems

I have a nonlinear, two point boundary value problem of the form
$F(x, y(x), y'(x); \Omega ) = y''$ along with some boundary conditions of the form $y_\Omega(0) = a_\Omega, y_{\Omega}(1) = b_\Omega$. ...

9
votes

2
answers

1k
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### Is there a differentiable but nonsmooth version of the continuous Implicit Function Theorem?

From the result discussed in Does the inverse function theorem hold for everywhere differentiable maps? (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obtain a ...

2
votes

2
answers

499
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### What is the closure of space of polynomials in a dense subspace along with a marked point equal to?

EDIT
Let $\mathbb{C}^{m*}$ be the space of non zero polynomials of degree at most
$d$ in two variables. So an element of this space is essentially
$$ f:=f_{00} + f_{10} x + f_{01} y + \ldots f_{0d}...

1
vote

1
answer

376
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### A version of implicit function theorem when sections are not everywhere smooth?

Let $V_1, V_2 \rightarrow M $ be smooth vector bundles over a manifold $M$ and $s_1: M \rightarrow V_1$
a smooth section transverse to the zero set and $s_2: M \rightarrow V_2$ a continuous section
...