Sparse effective membership problems via residue currents

We use residue currents on toric varieties to obtain bounds on the degrees of solutions to polynomial ideal membership problems. Our bounds depend on (the volume of) the Newton polytope of the polynomial system and are therefore well adjusted to sparse polynomial systems. We present sparse versions of Max N\"other's $AF+BG$ Theorem, Macaulay's Theorem, and Koll\'ar's Effective Nullstellensatz, as well as recent results by Hickel and Andersson-G\"otmark.