I have a family of single-variable analytic functions, $D(z)$, formed as follows.

Let $L$, $n$ and $T_{0}$ be positive integers, $\varphi_{1},\ldots, \varphi_{L}$ be analytic functions in ${\mathbb C}$, $p_{1},\ldots,p_{L}$ be polynomials in ${\mathbb C} \left[ z_{1},\ldots, z_{n} \right]$ of total degree at most $T_{0}$, $\zeta_{1},\ldots,\zeta_{L}$ be elements of ${\mathbb C}^{n}$ and $\theta_{1},\ldots,\theta_{n}$ be complex numbers.

Define
$$
f_{\lambda} \left( z_{1},\ldots, z_{n} \right)=p_{\lambda} \left( z_{1}, \ldots, z_{n} \right) \varphi_{\lambda} \left( \theta_{1}z_{1} + \cdots + \theta_{n} z_{n} \right).
$$
and
$$
D(z) = \det \left( f_{\lambda} \left( \zeta_{\mu} z \right) \right)_{1 \leq \lambda, \mu \leq L}.
$$

Here $\zeta_{\mu} z$ is scalar multiplication of $\zeta_{\mu}$ by $z$.

This is the notation from pages 192—193 of M. Waldschmidt's [1], which has the only result I have been able to find on the problem I am interested in (although my problem is slightly simpler than his, as I do not have his $\delta_{\mu,\lambda}$'s in the definition of my $D(z)$).

I am looking for a lower bounds for the zero multiplicity of $D(z)$ at $z=0$ that depends on $L$, $n$ and $T_{0}$ when $n \geq 2$. Although even such a result for just $n=2$ itself would be a big help.

If $n=1$, then it is known that the zero multiplicity must be at least $L(L-1)/2$ and this value is actually attained. So an analogous sharp result for $n \geq 2$ (but one that also depends on $n$ and $T_{0}$) would be great.

Waldschmidt has such a lower bound in his Lemma 7.2, but I have done quite a few examples and it seems that his Lemma 7.2 can be improved.

Does anyone have any references for such results?

Or suggestions for how to obtain such improved results?

What about a simplified case like where $\varphi_{1}(z)=\cdots=\varphi_{L}(z)=\exp(z)$?

Does that make the problem easier? Or something already known?

**Reference**

[1] Michel Waldschmidt, *Diophantine approximation on linear algebraic groups. Transcendence properties of the exponential function in several variables* (English), Grundlehren der Mathematischen Wissenschaften, 326, Berlin: Springer Verlag, pp. xxiii+633 (2000), ISBN: 3-540-66785-7, [MR1756786](https://mathscinet.ams.org/mathscinet-getitem?mr=MR1756786), [Zbl 0944.11024](https://zbmath.org/0944.11024).