Generically log smooth families via generators and relations

Let $f\text{\hspace{0.17em}⁣}:X\to {\mathbb{A}}_t^1$ be an affine flat morphism of finite type, and let $V=f^{-1}(0)$. Then, we obtain a morphism of log schemes $f\text{\hspace{0.17em}⁣}:(X\mathrm{\mid }V)\to ({\mathbb{A}}_t^1\mathrm{\mid }0)$. In this article, we develop algorithmic tools to study the log-geometric properties of $f$ by means of a presentation $$\mathrm{\Gamma }(X,{\mathcal{O}}_X)=\mathbb{k}[t,x_1,\dots ,x_n]\mathrm{/}(f_1,\dots ,f_r)\mathrm{.}$$ We obtain similar tools for projective flat morphisms when the homogeneous coordinate ring is given by generators and relations. We provide an implementation of our algorithms in Macaulay2. In a slightly different direction, we give some results on the sheaf ${\mathcal{L}\mathcal{S}}_V$ of log smooth structures on a toroidal crossing scheme $(V,\mathcal{P},\bar{\rho })$.