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We discuss the contributions to the Equation of State for the N$\ell$LOSkyrme pseudo-potential ($\ell$=2,3). We show that by adding 4th and 6th ordergradient terms, it is possible to fairly reproduce the spin/isospindecomposition of an equation of state obtained from \emph{ab-initio} methods.Moreover, by inspecting the partial-wave decomposition of the equation ofstate, we show for the first time a possible way to add explicit constraints onthe sign of the tensor terms of the Skyrme interaction.

Extended Skyrme pseudo-potential deduced from infinite matter properties