N.Y. Pub. Off. Law § 5
Every officer except a judicial officer, a notary public, a commissioner of deeds and an officer whose term is fixed by the constitution, having duly entered on the duties of his office, shall, unless the office shall terminate or be abolished, hold over and continue to discharge the duties of his office, after the expiration of the term for which he shall have been chosen, until his successor shall be chosen and qualified; but after the expiration of such term, the office shall be deemed vacant for the purpose of choosing his successor. An officer so holding over for one or more entire terms, shall, for the purpose of choosing his successor, be regarded as having been newly chosen for such terms. An appointment for a term shortened by reason of a predecessor holding over, shall be for the residue of the term only.
N.Y. Pub. Off. Law § 5