homogeneity condition $$\begin{gathered}
\label{eq:homogeneity for Fo}
E^\alpha\frac{\d F^o}{\d t^\alpha}+\frac{1-\delta}{2}s\frac{\d F^o}{\d s}=\frac{3-\delta}{2}F^o+D_\alpha t^\alpha+{\widetilde{D}}s+E,\quad\text{for some $D_\alpha,{\widetilde{D}},E\in{\mathbb C}$}.\end{gathered}$$ We see that the degree of the variable $s$ is determined by the conformal dimension of the Frobenius manifold. We will call a solution of the open WDVV equations homogeneous, if it satisfies condition .

In our paper we describe the space of homogeneous polynomial solutions of the open WDVV equations associated to the Frobenius manifolds of [*all*]{} finite irreducible Coxeter groups. In particular, this space is non-empty only for the Coxeter groups $A_N$, $B_N$ and $I_2(k)$.

Our approach to study solutions of the open WDVV equations is based on the following crucial observation of P. Rossi. Let $F = F(t^1,\ldots,t^N)$ be a Frobenius manifold potential and $F^o=F^o(t^1,\ldots,t^N,s)$ be a solution of the open WDVV equations satisfying . Then the $(N+1)$-tuple of functions $\left(\eta^{1\mu}\frac{\d F}{\d t^\mu},\ldots,\eta^{N\mu}\frac{\d F}{\d t^\mu},F^o\right)$ forms a [*vector potential*]{} of a [*flat F-manifold*]{}. This allows us to use the theory of flat F-manifolds to study solutions of the open WDVV equations.

Acknowledgements {#acknowledgements .unnumbered}
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We would like to thank Claus Hertling for useful discussions.

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the 