a fundamental role in the theory of Frobenius manifolds, because of the following result of C. Hertling, conjectured by B. Dubrovin [@Dub98]. Recall that a Frobenius manifold potential $F$ is called homogeneous, if there exists a vector field $E$ of the form $$\begin{gathered}
\label{eq:Euler field for Frobenius}
E=\sum_{\alpha=1}^N\underbrace{(q_\alpha t^\alpha+r^\alpha)}_{=:E^\alpha}\frac{\d}{\d t^\alpha},\quad q_\alpha,r^\alpha\in{\mathbb C},\quad q_1=1,\end{gathered}$$ such that $$\begin{gathered}
E(F)=E^\alpha\frac{\d F}{\d t^\alpha}=(3-\delta)F+\frac{1}{2}A_{\alpha\beta}t^\alpha t^\beta+B_\alpha t^\alpha+C,\quad\text{for some $\delta,A_{\alpha\beta},B_\alpha,C\in{\mathbb C}$}.\end{gathered}$$ The number $\delta$ is called the [*conformal dimension*]{} and the vector field $E$ is called the [*Euler vector field*]{}. C. Hertling proved that any generically semisimple Frobenius manifold (see Section \[subsection:flat F-manifolds\] for definition), whose potential is polynomial $F \in {{\mathbb{C}}}[t^1,\dots,t^N]$ and homogeneous with the Euler vector field of the form $E=q_\alpha t^\alpha\frac{\d}{\d t^\alpha}$, where $q_\alpha>0$, can be expressed as the product of the Frobenius manifolds corresponding to finite irreducible Coxeter groups [@Hert02 Theorem 5.25].

In the second part of the paper we study the space of polynomial solutions of the open WDVV equations associated to the Frobenius manifolds of finite irreducible Coxeter groups. Note that all solutions of the open WDVV equations, considered in the works [@HS12; @PST14; @BCT18; @BCT19], are associated to a homogeneous Frobenius potential $F$ and, moreover, the function $F^o$ satisfies the 