Are you talking about the Orange Flesh sweet potatoes (OFSP) or normal sweet potatoes? Because OFSP is a genetically improved variety that contain high level of beta-carotene sometimes comparable to carrots.
But talking of a nutrient inducing the biosynthesis of beta-carotene in normal potatoes!! That may be a research topic! Though some mineral elements are Always needed as enzymes cofactors during synthesis, but then the pathway should exist.
Yellow sweet potato is Rich in retinal, fats help in biosynthesis, since it also contain sugar, riboflavin will aid in it's biosynthesis together with niacin
Orange flesh and skin sweet potatoes are genetically modified with Beta Carotene . The main reason for this fortification is to aid in improving non heme iron absorption.