Learn Polish: Kiedy piekę ciasto, często nie mam jajek. - When I bake a cake, I often have no eggs.

Jajko (egg) is a neuter noun. To form the Genitive plural of neuter (and feminine) nouns, you usually drop the final vowel. But dropping the -o from jajko leaves jajk, which ends in a difficult consonant cluster (-jk). To make it easier to pronounce, Polish inserts a "movable e" between the consonants, giving us jajek.

Why do we use the Genitive case for jajek in the first place?

It is because of the phrase nie mam (I don't have). The verb mieć (to have) usually takes the Accusative case, but when you make it negative, it forces the following noun into the Genitive case. This is a strict rule in Polish: a negated Accusative object always becomes Genitive.

If negative verbs trigger the Genitive, why is ciasto (cake) just in its standard dictionary form?

Ciasto is the object of the verb piekę (I bake). Since piekę is an affirmative (positive) verb, it takes the Accusative case. For neuter nouns like ciasto, the Accusative looks exactly the same as the Nominative (dictionary) form. The Genitive rule here only applies to the object of the negative verb nie mam.

Piekę means "I bake", but what is the infinitive (dictionary) form of this verb?