Problem Solving Set 11

14 July 2012

1. Call a polynomial P (x_{1}, …, x_{k}) good if there exist 2 × 2 real matrices A_{1}, …, A_{k} such that

P (x_{1}, …, x_{k}) = det (∑_{i=1}^{k} x_{i}A_{i}) .

Find all values of k for which all homogeneous polynomials with k variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)

2. Show that the equation

2x^{4} + 1 = y^{2}

has no integer solution.