Laws of Inertia in Higher Degree Binary Forms

If there are a positive (and b negative) coefficients among the λj’s, then we say that (1.1) has badge (a, b). Two linear forms are distinct if they (and their 2s-th powers) are not proportional; a representation is honest if its summands are pairwise distinct. Let B(p) denote the set of possible badges of honest representations of p. Badges are ordered componentwise: (a, b) (c, d) if a ≤ c and b ≤ d. Any minimal badge under this ordering is called a signature of p. The set of signatures of p is denoted S(p); in case p has a unique signature, we call it the signature of p and denote it S(p). The length of p is the minimum value of a+ b in B(p); it is the shortest possible representation (1.1). If q is obtained from p by an invertible linear change of variables, then B(q) = B(p). Identities such as x + (x+ y) + y = (x− y) + (x+ 2y) − 3y = 2(x+ 1 2 y) + 3 2 y