Generalisation of Scott permanent identity

Let x = {x1, . . . , xr}, y = {y1, . . . , yn}, z = {z1, . . . , zn} be three sets of indeterminates. We give the value of the determinant ∣ ∣ ∣ ∣ ∣ ∏ x∈x (xy − z) ∣ ∣ ∣ ∣ ∣ y∈y,z∈z when specializing y and z to the set of roots of yn − 1 and zn − ξn respectively. In the case where r = 2 and x = {1, 1} the determinant ∣ ∣(y− z) ∣ ∣ y∈y,z∈z factorizes into the determinant of the Cauchy matrix [(y − z)] and its permanent. Scott [10, 8] found the value of this permanent when specializing y to the roots of y − 1 and z to the roots of z +1. Han [3] described more generally the case where z is the set of roots of z+az + b instead of z+1. Instead of restricting to r = 2 and specializing x, we shall consider the determinant ∣ ∣ ∣ ∣ ∣ ∏ x∈x (xy − z) ∣ ∣ ∣ ∣ ∣ y∈y,z∈z , and obtain in Th. 2 its value when specializing y and z. The remarkable feature is that this value is a product of sums of monomial functions in x without multiplicities, thus extending the factorized expressions of [10, 3]. ∗partially supported by the ANR project MARS (BLAN06-2 134516).