Posterior superior alveolar artery

The posterior superior alveolar artery (posterior dental artery) is a branch of the maxillary artery.  It is one of two or three superior alveolar arteries. It provides arterial suply to the molar and premolar teeth, maxillary sinus and adjacent bone, and the gingiva.

Origin
The artery typically arises from maxillary artery within the pterygopalatine fossa. It frequently arises in conjunction with the infraorbital artery.

Course
It passes inferior-ward upon the infratemporal surface of maxilla before ramifying.

Branches
It emits branches that pass through foramina on the posterior aspect of the maxilla alongside the posterior superior alveolar nerves. 

Some branches enter the alveolar canals to supply the upper molar and premolar teeth as well as the maxillary sinus and adjacent bone.

Some branches pass anterior-ward across the alveolar process to supply the gingiva.