Corollary of the Malgrange Preparation Theorem

Too long for a comment.

I found mentioned Russian translation "Особенности дифференцируемых отображений" of Malgrange’s paper. Although Russian is my native language (so I can translate for you a few places of the paper, if needed) unfortunately, I am far from the topic of the paper and don’t understand it.

I see the formula very similar (5) neither at p. 185 nor at other pages (183, 184, 186, 187, 188, 188).

Here is the mentioned corollary:

Here is its translation.

Let shoots [Sorry, I don’t know the right translation of this word. A main meaning of a russian word “ростки” is sprouts. AR] $\varphi_1,\dots,\varphi_p\in \mathcal E(x)$. Then the following conditions are equivalent:

($a$)’ $\varphi_1,\dots,\varphi_p$ generate $\mathcal E(x)$ as a $\mathcal E(y)$-module.

($\hat a$)’ $\varphi_1,\dots,\hat\varphi_p$ generate $\hat{\mathcal E}(x)$ as a $\hat{\mathcal E}(y)$-module.

($b$)’ images $\varphi_1,\dots,\varphi_p$ in a ring $\mathcal E(x)/ \mathcal E(x)u^*{\frak m}({\mathcal E}(y))$ generate it as a linear space over $\Bbb R$.

($\hat b$)’ images $\hat\varphi_1,\dots, \hat\varphi_p$ in a ring $\hat{\mathcal E}(x)/\hat{\mathcal E}(x) \hat u^*{\frak m}(\hat{\mathcal E}(y))$ generate it as a linear space over $\Bbb R$.

Example 2. Weierstraß’ prepartion theorem. Let $F(x_1,\dots, x_n)\in\mathcal E_n$ be a regular shoot of order $p$ with respect to $x_n$ (that is $F(0,\dots, 0, x_n)$) has in a point $x_n=0$ zero of order exactly $p$). Moreover, let again $m=n$ and a shoot of a map $u$ is defined as

$$(x_1,\dots, x_{n_1}, x_n)\to (x_1,\dots, x_{n-1}, F(x_1,\dots, x_n)).$$

Obviuously, an ideal generated in the ring $\hat{\mathcal E}_n$ by elements $x_1,\dots, x_{n-1}, \hat F$ coincides with an ideal generated in this ring by elements $x_1,\dots, x_{n-1}, x_n^p$. Now we can apply the corollary from theorem 1 (we need the equivalency of conditions ($a$)’ and ($\hat b$)’), putting $\varphi_i=x^{p-i}_n$ ($1\le i\le p$). In other words, each shoot $f\in\mathcal E_n$ can be written in the form

$$f(x_1,\dots, x_n)=\sum_{i=1}^p g_i(x_1,\dots, x_{n-1}, F)x^{p-i}_n,$$

where all $g_i\in\mathcal E_n$. Introduce a notation $h_i(x_1,\dots, x_{n-1})=g_i(x_1,\dots, x_{n-1},0)$ and remark that $g_i-h_i=x_nk_i$ for some $k_i\in\mathcal E_n$. Substituting $x_n$ by $F$, we obtain the following result:

(W) Let $F\in\mathcal E_n$ be a regular shoot of order $p$ with respect to $x_n$. Than for any shoot $f\in\mathcal E_n$ there exist shoots $Q\in\mathcal E_n$ and $h_i\in\mathcal E_{n-1}$ such that

$$f(x_1,\dots, x_n)=F(x_1,\dots, x_n)Q(x_1,\dots, x_n)+\sum_{i=1}^p h_i(x_1,\dots, x_{n-1})x^{p-i}_n.$$

Exactly the same claim for analytical functions constitutes a Weierstraß’ preparation theorem in Рюккерт’s form. The same theorem in Weierstraß’ form is obtained applying (W) to the case $f=x^p_n$;...

Even this is too long for a comment, and also should be intended as an (hopefully useful) addendum to the previous answer of Alex Ravsky.

The paper [3] cited by Vishik is the Russian translation of the proceeding paper [2]: its Zentralblatt MATH review state that it is simply a commented summary of the paper [1], appeared earlier in the Seminaire Henri Cartan, so we should look at these papers in order to seek for a proof of the corollary which Vishik cites. However, even looking at those papers and at reference [4], the result cited by Vishik is not clearly stated in the form reported in his paper.

But let's have a look at the division theorem stated at p. 188 of [3] as formula [A], formerly proved as a cornerstone in the theory of PDEs by Lars Hörmander (division by a polynomial) and by Stanisław Łojasiewicz (division by a real analytic function) and obtained by Malgrange as a corollary of its even more general preparation theorem (see also [1], paper IV, p. 22-1 and [4], chapter IX, §2 pp. 185-186): it states that, given a (generalized) polynomial in the variable $t$ (here and below I follow the question notation, not the ones of the references cited, and by $\mathscr{E}_{j}$ I mean the space of germs of smooth functions in $j$ variables) $$ \Pi(t,x,a)=t^{k+1}+\sum_{i=1}^k a_i(x)\cdot t^i\in\mathscr{E}_{n+k+1}\quad a=(a_1,\ldots,a_k) $$ and a germ $f\in\mathscr{E}_n$, there exists a germ $q\in\mathscr{E}_{n+k+1}$ and germs $h_i\in\mathscr{E}_{n+k}$, $i=1,\ldots,k$ such that: $$ f(t,x)=\Pi(t,x,a)q(t,x,a)+\sum_{i=1}^k h_i(x,a)\cdot t^i $$ This implies that $$ b(x)\triangleq \frac{f(x)-\sum_{i=1}^k h_i(t,x,a)\cdot t^i}{q(t,x,a)}\in\mathscr{E}_{n+k+1} $$ i.e. $b$ being again a smooth germ since by construction it is divisible by $q$, thus $$ \Pi(t,x,a)=t^{k+1}+\sum_{i=1}^k a_i(x)\cdot t^i=b(x) $$ If we consider the change of variables used by Vishik in order to describe the set $Q\cap U$ $$ (t,x_1,\ldots,x_{n-1}, x_n)\mapsto(t,x_1,\ldots,x_{n-1}, \varphi(t,x)), $$ the sought for result is derived from the division theorem, i.e. $$ t^{k+1}+\sum_{i=1}^k a_i(x_1,\ldots,x_{n-1}, \varphi(t,x))\cdot t^i = b(x_1,\ldots,x_{n-1}, \varphi(t,x)) $$ Finally, I want to share with all the follow notes

1. In previous revisions of this "answer" (a batter name would be long comment) I somewhat messed with the notation for germs: this is due to the fact that in my in main reference [4] for such topics, both Malgrange's preparation theorem and the division theorem above are proved globally for functions in $\mathscr{E}\equiv C^\infty$ (following the notation introduced for the theory of distributions by L. Schwartz). The domain of the functions for which the division problem is solved in [4] is assumed not only to be a subset of the standard Euclidean space $\Bbb R^n$ but also a submanifold of a smooth $C^\infty$ manifold $X$.
2. Following what I stated in the previous point, I feel right to give an advice: perhaps the best place to learn about the results of Malgrange is the book [4], because the approach adopted by Tougeron follows the approach to the problem proposed by S. Łojasiewicz, who considerably simplified Malgrange's proof. See the historical *remarque 2.9" in [4], chapter IX, §2 p. 187 for a concise description of various approaches. However, despite their importance in analysis, in my opinion, it's quite difficult for an analyst to master those techniques by using simply their standard curriculum preparation.

[1] Bernard Malgrange (1962-1964), "Le théorème de préparation en géométrie différentiable. I: Position du problème (MR160234). II: Rappels sur les fonctions différentiables (MR160235). III: Propriétés différentiables des ensembles analytiques (MR160236). IV: Fin de la démonstration (MR160237)" (French). Topologie Différentielle, Séminaire Henri Cartan, tome 15 (1962/63), No. 11, 14 pp. (1964), No. 12, 9 pp. (1964), No. 13, 12 pp. (1964), No. 22, 8 pp. (1964), Zbl 0119.28501.

[2] Bernard Malgrange (1964), "The preparation theorem for differentiable functions" (English), in M.F. Atiyah et al. (Ed.) Differential Analysis, Bombay Colloquium 1964, Tata Studies in Mathematics 2, London: Oxford University Press, pp. 203-208, MR0182695, Zbl 0137.03601.

[3] Bernard Malgrange (1968), "The preparation theorem for differentiable functions" (Russian), Osobennosti differentsiruemykh Otobrazhenij, 183-189, Zbl 0199.37903.

[4] Jean-Claude Tougeron (1972), Ideaux de fonctions différentiables (French) Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 71. Berlin-Heidelberg-New York’ Springer-Verlag. pp. VII+219, MR0440598, Zbl 0251.58001.