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

Let $L$ and $n$ be positive integers, $\varphi_{1},\ldots, \varphi_{L}$ be analytic functions in ${\mathbb C}$, $p_{1},\ldots,p_{L}$ polynomials in ${\mathbb C} \left[ z_{1},\ldots, z_{n} \right]$, $\zeta_{1},\ldots,\zeta_{L}$ 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 $$ D(z) = \det \left( f_{\lambda} \left( \zeta_{\mu} z \right) \right)_{1 \leq \lambda, \mu \leq L}. $$

This is the notation from pages 192--193 of M. Waldschmidt's book ``Diophantine Approximation on Linear Algebraic Groups'', 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 $z=0$ when $n \geq 2$. Although even 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 $n \geq 2$ would be great.

Waldschmidt has 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?