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keep under (something)
1. To remain beneath something, especially so as to hide. I kept under the covers until the scary sound of thunder went away. Kids in the '50s and '60s were instructed to keep under their desks in the event of a nuclear attack.
2. To store or hide someone or something beneath something. In this usage, a noun or pronoun is used between "keep" and "under." He tipped off the police that Tommy the Tooth was keeping the drug stash under the floorboards in his kitchen. The crew member was caught keeping a stowaway under a blanket in a storage compartment of the ship. My grandmother kept her life's savings under her mattress, claiming that she didn't trust the banks.
3. To cause someone or some group to continue to undergo or receive the effect of some condition or action. In this usage, a noun or pronoun is used between "keep" and "under." Authorities have kept the man under surveillance for his extremist political views. You have to keep the kids under constant supervision. The government has kept the population under control with the constant threat of violence.
keep someone or something under something
to store, hide, or cache someone or something beneath something. Keep Max under the packing crate until the police go away. I keep a box of extra sweaters under the bed.
keep under something
to remain beneath something. Keep under the packing crate, Max. I ordered the dog to keep under the table.