  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.
