Remark on Arnold's problem concerning quaternionic determinants S.Duzhin 16-Oct-98 This note is related to Problem 2 of the September (1998) list of Arnold's problems. We prove that for any n the determinant of the complexification of a quaternionic matrix nxn can be represented as a sum of 4 squares of rational functions in the real components of the matrix elements, but for n>1, it cannot be represented as a sum of squares of polynomials. Let A be a quaternionic matrix nxn. Replacing each element by the corresponding 2x2 complex matrix, we will get a 2nx2n complex matrix CA. This construction provides an isomorphism of the algebra Mat(n,H) onto a subalgebra of Mat(2n,C). Using the multiplication by elementary matrices belonging to the subalgebra, we can reduce the matrix CA to a triangular matrix with the same determinant. The determinant of the triangular matrix is equal to the product of n sums of 4 squares, each corresponding to the i-th 2x2 block. This product is representable as a sum of 4 squares by Lagrange identity. Therefore, we have: Theorem 1. The determinant of CA is a sum of squares of 4 rational functions. Remark. This representation is by far not unique. Example. For n=2, possible explicite formulas are: det(CA) = (f1/d)^2 + (f2/d)^2 + (f3/d)^2 + (f4/d)^2, where the denominator d = p1^2+p2^2+p3^2+p4^2, and the numerators are f1 = p1^3*s1-p1^2*p2*s2-p1^2*p3*s3-p1^2*p4*s4-p1^2*q1*r1+p1^2*q2*r2 +p1^2*q3*r3+p1^2*q4*r4+p1*p2^2*s1-2*p1*p2*q3*r4+2*p1*p2*q4*r3+p1*p3^2*s1 +2*p1*p3*q2*r4-2*p1*p3*q4*r2+p1*p4^2*s1-2*p1*p4*q2*r3+2*p1*p4*q3*r2 -p2^3*s2-p2^2*p3*s3-p2^2*p4*s4-p2^2*q1*r1+p2^2*q2*r2-p2^2*q3*r3 -p2^2*q4*r4-p2*p3^2*s2+2*p2*p3*q2*r3+2*p2*p3*q3*r2-p2*p4^2*s2 +2*p2*p4*q2*r4+2*p2*p4*q4*r2-p3^3*s3-p3^2*p4*s4-p3^2*q1*r1-p3^2*q2*r2 +p3^2*q3*r3-p3^2*q4*r4-p3*p4^2*s3+2*p3*p4*q3*r4+2*p3*p4*q4*r3-p4^3*s4 -p4^2*q1*r1-p4^2*q2*r2-p4^2*q3*r3+p4^2*q4*r4 f2 = p1^3*s2+p1^2*p2*s1+p1^2*p3*s4-p1^2*p4*s3-p1^2*q1*r2-p1^2*q2*r1 +p1^2*q3*r4-p1^2*q4*r3+p1*p2^2*s2+2*p1*p2*q3*r3+2*p1*p2*q4*r4+p1*p3^2*s2 -2*p1*p3*q1*r4-2*p1*p3*q3*r2+p1*p4^2*s2+2*p1*p4*q1*r3-2*p1*p4*q4*r2 +p2^3*s1+p2^2*p3*s4-p2^2*p4*s3-p2^2*q1*r2-p2^2*q2*r1-p2^2*q3*r4 +p2^2*q4*r3+p2*p3^2*s1-2*p2*p3*q1*r3-2*p2*p3*q4*r2+p2*p4^2*s1 -2*p2*p4*q1*r4+2*p2*p4*q3*r2+p3^3*s4-p3^2*p4*s3+p3^2*q1*r2-p3^2*q2*r1 -p3^2*q3*r4-p3^2*q4*r3+p3*p4^2*s4+2*p3*p4*q3*r3-2*p3*p4*q4*r4-p4^3*s3 +p4^2*q1*r2-p4^2*q2*r1+p4^2*q3*r4+p4^2*q4*r3 f3 = p1^3*s3-p1^2*p2*s4+p1^2*p3*s1+p1^2*p4*s2-p1^2*q1*r3-p1^2*q2*r4 -p1^2*q3*r1+p1^2*q4*r2+p1*p2^2*s3+2*p1*p2*q1*r4-2*p1*p2*q2*r3+p1*p3^2*s3 +2*p1*p3*q2*r2+2*p1*p3*q4*r4+p1*p4^2*s3-2*p1*p4*q1*r2-2*p1*p4*q4*r3 -p2^3*s4+p2^2*p3*s1+p2^2*p4*s2+p2^2*q1*r3+p2^2*q2*r4-p2^2*q3*r1 +p2^2*q4*r2-p2*p3^2*s4-2*p2*p3*q1*r2+2*p2*p3*q4*r3-p2*p4^2*s4 -2*p2*p4*q2*r2+2*p2*p4*q4*r4+p3^3*s1+p3^2*p4*s2-p3^2*q1*r3+p3^2*q2*r4 -p3^2*q3*r1-p3^2*q4*r2+p3*p4^2*s1-2*p3*p4*q1*r4-2*p3*p4*q2*r3+p4^3*s2 +p4^2*q1*r3-p4^2*q2*r4-p4^2*q3*r1-p4^2*q4*r2 f4 = p1^3*s4+p1^2*p2*s3-p1^2*p3*s2+p1^2*p4*s1-p1^2*q1*r4+p1^2*q2*r3 -p1^2*q3*r2-p1^2*q4*r1+p1*p2^2*s4-2*p1*p2*q1*r3-2*p1*p2*q2*r4+p1*p3^2*s4 +2*p1*p3*q1*r2-2*p1*p3*q3*r4+p1*p4^2*s4+2*p1*p4*q2*r2+2*p1*p4*q3*r3 +p2^3*s3-p2^2*p3*s2+p2^2*p4*s1+p2^2*q1*r4-p2^2*q2*r3-p2^2*q3*r2 -p2^2*q4*r1+p2*p3^2*s3+2*p2*p3*q2*r2-2*p2*p3*q3*r3+p2*p4^2*s3 -2*p2*p4*q1*r2-2*p2*p4*q3*r4-p3^3*s2+p3^2*p4*s1+p3^2*q1*r4+p3^2*q2*r3 +p3^2*q3*r2-p3^2*q4*r1-p3*p4^2*s2-2*p3*p4*q1*r3+2*p3*p4*q2*r4+p4^3*s1 -p4^2*q1*r4-p4^2*q2*r3+p4^2*q3*r2-p4^2*q4*r1 The second question is whether the determinant of CA can be written as a sum of squares of polynomials. Let p q r s be a quaternionic 2x2 matrix A. The corresponding 4x4 complex matrix is p1 + I p2 p3 + I p4 q1 + I q2 q3 + I q4 -p3 + I p4 p1 - I p2 -q3 + I q4 q1 - I q2 r1 + I r2 r3 + I r4 s1 + I s2 s3 + I s4 -r3 + I r4 r1 - I r2 -s3 + I s4 s1 - I s2 whose determinant is the following homogeneous polynomial F of degree 4 in 16 real variables p_i, q_i, r_i, s_i: F=p1^2*s1^2+p1^2*s2^2+p1^2*s3^2+p1^2*s4^2-2*p1*q1*r1*s1-2*p1*q1*r2*s2 -2*p1*q1*r3*s3-2*p1*q1*r4*s4-2*p1*q2*r1*s2+2*p1*q2*r2*s1+2*p1*q2*r3*s4 -2*p1*q2*r4*s3-2*p1*q3*r1*s3-2*p1*q3*r2*s4+2*p1*q3*r3*s1+2*p1*q3*r4*s2 -2*p1*q4*r1*s4+2*p1*q4*r2*s3-2*p1*q4*r3*s2+2*p1*q4*r4*s1+p2^2*s1^2+ p2^2*s2^2+p2^2*s3^2+p2^2*s4^2+2*p2*q1*r1*s2-2*p2*q1*r2*s1-2*p2*q1*r3*s4 +2*p2*q1*r4*s3-2*p2*q2*r1*s1-2*p2*q2*r2*s2-2*p2*q2*r3*s3-2*p2*q2*r4*s4 +2*p2*q3*r1*s4-2*p2*q3*r2*s3+2*p2*q3*r3*s2-2*p2*q3*r4*s1-2*p2*q4*r1*s3 -2*p2*q4*r2*s4+2*p2*q4*r3*s1+2*p2*q4*r4*s2+p3^2*s1^2+p3^2*s2^2+p3^2*s3^2 +p3^2*s4^2+2*p3*q1*r1*s3+2*p3*q1*r2*s4-2*p3*q1*r3*s1-2*p3*q1*r4*s2 -2*p3*q2*r1*s4+2*p3*q2*r2*s3-2*p3*q2*r3*s2+2*p3*q2*r4*s1-2*p3*q3*r1*s1 -2*p3*q3*r2*s2-2*p3*q3*r3*s3-2*p3*q3*r4*s4+2*p3*q4*r1*s2-2*p3*q4*r2*s1 -2*p3*q4*r3*s4+2*p3*q4*r4*s3+p4^2*s1^2+p4^2*s2^2+p4^2*s3^2+p4^2*s4^2 +2*p4*q1*r1*s4-2*p4*q1*r2*s3+2*p4*q1*r3*s2-2*p4*q1*r4*s1+2*p4*q2*r1*s3 +2*p4*q2*r2*s4-2*p4*q2*r3*s1-2*p4*q2*r4*s2-2*p4*q3*r1*s2+2*p4*q3*r2*s1 +2*p4*q3*r3*s4-2*p4*q3*r4*s3-2*p4*q4*r1*s1-2*p4*q4*r2*s2-2*p4*q4*r3*s3 -2*p4*q4*r4*s4+q1^2*r1^2+q1^2*r2^2+q1^2*r3^2+q1^2*r4^2+q2^2*r1^2 +q2^2*r2^2+q2^2*r3^2+q2^2*r4^2+q3^2*r1^2+q3^2*r2^2+q3^2*r3^2+q3^2*r4^2 +q4^2*r1^2+q4^2*r2^2+q4^2*r3^2+q4^2*r4^2 Theorem 2. The polynomial F cannot be represented as the sum of squares of any number of real polynomials. Proof. Note that all the monomials in F which are squares, have the form (p_i*s_j)^2 or (q_i*r_j)^2. Suppose that F = F_1^2 + ... + F_n^2. Then: 1. Polynomials F_i are homogeneous of degree 2 in the 16 variables p_i, q_i, r_i, s_i. 2. Polynomials F_i do not contain squares of the variables. 3. Polynomials F_i do not contain products of variables other than p_i*s_j and q_i*r_j. 4. Therefore they are linear forms in the 32 variables x_{ij}=p_i*s_j and y_{ij}=q_i*r_j. 5. In the new variables x_{ij} and y_{ij} the initial polynomial F is a quadratic form of signature 0 (it has 16 pluses and 16 minuses). Therefore it cannot be equal to the sum of squares of linear forms. Corollary. For any n>1 the determinant of the complexified matrix CA cannot be represented as a sum of squares of real polynomials.