On the tropicalization of D-ideals

Following the Gröbner basis theoretic study of tropical geometry due to Maclagan and Sturmfels, we construct a notion of tropicalization for ideals in various rings of differential operators. We first treat the classical case, tropicalizing left ideals in the ring of differential operators with analytic or formal coefficients. Our construction is benefited from the recent development of Gröbner basis theory in this setting. Most of our elaboration will focus on the tropicalization of left ideals in the Weyl algebra over a characteristic 0 field with valuation. In order to achieve this, the relevant Gröbner basis theory in this generality is developed. In both cases, we will prove the existence of a tropical basis and relate the tropicalization of a D-ideal with the usual tropicalization of its characteristic varieties.