Some determinants involving binary forms

In this paper, we study arithmetic properties of certain determinants involving powers of i²+cij+dj², where c and d are integers. For example, for any odd integer $n>1$ with ( dn)=-1 we prove that [ (i²+cij+dj²/n)]0≤ i,j≤ n-1 is divisible by φ(n)², where (·/n) is the Jacobi symbol and φ is Euler's totient function. This confirms a previous conjecture of the second author.