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Motivated by work of Dragt and Abell on accelerator physics, we study thecompletion of symplectic jets by polynomial maps of low degrees. We useAnders\'en-Lempert Theory to prove that symplectic completions always exist,and we prove the degree bound conjectured by Dragt and Abell in the physicallyrelevant cases. However, we disprove the degree bound for 3-jets in dimension4. This follows from the fact that if $\Sigma$ is the disjoint union of $r=7$involutive lines in $\mathbb P^3$, then $\Sigma$ is contained in a degree $d=4$hypersurface, i.e., the restriction morphism $\iota:H^0(\mathbb P^3,\mathcalO(4))\rightarrow H^0(\Sigma,\mathcal O(4))$ has a nontrivial kernel (Todd). Wegive two new proofs of this fact, and finally we show that if $(r,d)\neq (7,4)$then the map $\iota$ has maximal rank.

Polynomial Completion of Symplectic Jets and Surfaces Containing Involutive Lines