Abstract: Given an ideal $J$, its $n^{th}$ symbolic power $J^{(n)}$, morally speaking, consists of those polynomials which vanish to order $n$ along the variety defined by $J$, which we'll call $V(J)$. A natural question to ask is what is the minimum degree of a polynomial which vanishes to order $n$ along $V(J)$; equivalently one can ask for the minimum degree of a polynomial in the ideal $J^{(n)}$. We will call this minimum degree $a(J^{(n)})$. For instance, there is a (still unsolved) question of Nagata about the minimum degree of a polynomial which vanishes to order $n$ at a set of generic points in the plane. We will introduce symbolic powers, the invariant $a(J^{(n)})$, and then discuss how this problem can be approached for squarefree monomial ideals.
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