Determinantal complexities and field extensions

Let double-struck F be a field of characteristic ≠ 2. The determinantal complexity of a polynomial P ∈ double-struck F[x₁,..., x _n] is defined as the smallest size of a matrix M whose entries are linear polynomials of x_i 's over double-struck F, such that P = det(M) as polynomials in double-struck F[x₁,..., x_n]. To determine the determinantal complexity of the permanent polynomial is a long-standing open problem. Let double-struck K be an extension field of double-struck F; then P can be viewed as a polynomial over double-struck K. We are interested in the comparison between the determinantal complexity of P over double-struck K (denoted as dc_{double-struck K}(P))), and that of P over double-struck F (denoted as dc_{double-struck F}(P). It is clear that dc_{double-struck K}(P) ≤, dc_{double-struck F}(P) and the question is whether strict inequality can happen. In this note we consider polynomials defined over ℚ. For P = x₁² + ⋯ + x_n², there exists a constant multiplicative gap between dc_ℝ(P) and dc_ℂ(P): we prove dc _ℝ(P) ≥ n while ⌈n/2⌉ + 1 ≥ dc _ℂ(P). We also consider additive constant gaps: (1) there exists a quadratic polynomial Q ∈ ℚ [x, y], such that dc_ℚ-(Q) = 3 and ; (2) there exists a cubic polynomial C ∈ ℚ[x, y] with a rational zero, such that dc_ℚ(C) = 4 and dc_ℚ-(C) = 3. For additive constant gaps, geometric criteria are presented to decide when dc_ℚ = dc_ℚ-.