Minimal rank of primitively n-universal integral quadratic forms over local rings

Abstract Let F be a local field and let R be its ring of integers. For a positive integer n , an integral quadratic form defined over R is called primitively n -universal if it primitively represents all quadratic forms of rank n . It was proved in [7] that the minimal rank of primitively 1-universal quadratic forms over the p -adic integer ring $$\mathbb {Z}_p$$ Z p is 2 if p is odd, and 3 otherwise. In this article, we completely determine the minimal rank of primitively n -universal quadratic forms over R for any positive integer n and any local ring R such that 2 is a unit or a prime.