SetDirectory["~/KappaLib/"];
<<kappaLib-1.2.m

Verify that the Fresnel polynomial for the QDCM medium class in 

   Lindell, Bergamin, Favaro : Decomposable medium conditions in four - dimensional representation
   
factorises into the product of two quadratic forms.

Define most general QDCM medium

QQ=emMatrix["q",4,Structure->"General"];
CC=emMatrix["a",4,Structure->"AntiSymmetric"];
DD=emMatrix["b",4,Structure->"AntiSymmetric"];
(* rho1 = scalar density of weight 1 *)
(* rho2 = scalar density of weight 1 *)
(* C1 = constant *)

kappa=C1 emIdentityKappa[]+emQMedium[rho1,QQ] + emBiProduct[rho2,CC,DD];

Compute Fresnel polynomial

vars={x0,x1,x2,x3};
fresnel=emKappaToFresnel[kappa,vars];

Check that the Fresnel polynomial factorises into a product if two quadratic forms

(* define adjugate of Q *)
adjQ = Table[
			(-1)^(i+j)Det[Drop[QQ,{i,i},{j,j}]],
			{j,1,4},{i,1,4}
			];

(* check *)
Simplify[adjQ/Det[QQ]-Inverse[QQ]]

factor1=vars.(rho1 Det[QQ]QQ +2 rho2 CC.adjQ.Transpose[DD]).vars;
factor2=vars.QQ.vars;
fresnelExp= rho1^2factor1 factor2;

Simplify[fresnel-fresnelExp]

Note 1: The above decomposition is valid also when Q is singular.

Note 2: The factor 2 in front of rho2 does not appear in the original paper.