4-Generated Pseudo Symmetric Monomial Curves with not Cohen-Macaulay Tangent Cones

In this article, standard bases of some toric ideals associated to 4-generated pseudo symmetric semigroups with not Cohen-Macaulay tangent cones at the origin are computed. As the tangent cones are not Cohen-Macaulay, non-decreasingness of the Hilbert function of the local ring was not guaranteed. Therefore, using these standard bases, Hilbert functions are explicitly computed as a step towards the characterization of Hilbert function. In addition, when the smallest integer satisfying $k({\alpha }_2+1)<(k-1){\alpha }_1+(k+1){\alpha }_{21}+{\alpha }_3$ is $1$, it is proved that the Hilbert function of the local ring is non-decreasing.