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?

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, Zbl 0944.11024.