Symplectic $T_7$, $T_8$ singularities and Lagrangian tangency orders

We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic $T_7$ singularities. We define discrete symplectic invariants - the Lagrangian tangency orders. We use these invariants to distinguish symplectic singularities of classical $A-D-E$ singularities of planar curves, $S_5$ singularity and $T_7$ singularity. We also give the geometric description of these symplectic singularities.

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