According to the Oregon Court of Appeals, No!
Last month the Court decided a rather unique case in State v. Sanchez-Anderson. A Beaverton officer saw a rental vehicle with out-of-state license plates. The Defendant in this case was passenger of this vehicle. After being stopped for a minor traffic violation, the officer let the Defendant (passenger) walk away as there is almost never a lawful basis to seize the passenger in a traffic stop. (It's OK to walk away!)
The Officer found various drugs and items with the driver and in the driver's vehicle. The one item of interest to the Defendant was a single 'used' syringe found under the passenger's seat. Based on Driver's possession and this single syringe the Defendant was arrested and later searched for 'possession of controlled substances'.
Under Oregon law Probable Cause exists, "only if the arresting officer subjectively believes it is more likely than not that an offense has been committed and that belief is objectively reasonable." State v. Miller, 345 Or 176, 184, 191 P3d 651 (2008). To make this determination it is necessary to look at the 'totality of the circumstances'. Generally speaking, drug possession can be established through proof of either actual or constructive possession of a specific controlled substance. An officer has objective probable cause to believe that a person constructively possesses contraband if the circumstances show that, more likely than not, the defendant knowingly exercises control over the contraband or has the right to do so. Oregon law requires that, "some facts link the defendant’s presence in the vehicle where the drugs were observed to the defendant’s right to control those drugs."
For all these reasons, the syringe found under the Defendant's seat, was never in her 'possession' in the Court's view. There was no evidence of her use of the syringe, DNA analysis, fingerprints etc. Under the totality of the circumstances she never actually or constructively possessed the syringe.
So to answer the question, "do you possess the contraband under your seat?" At least as it applies to the facts of this case, nope!