Tight bounds for rational sums of squares over totally real fields

摘要

Let K be a totally real Galois number field. Hillar proved that if f ∈ ℚ[x ₁, ..., x _n ] is a sum of m squares in K[x ₁, ..., x _n ], then f is a sum of N(m) squares in ℚ[x ₁, ..., x _n ], where N(m) ≤ 2[K:ℚ]+1 · $nleft( {_2^{[K:mathbb{Q}] + 1} } right)n$nleft( {_2^{[K:mathbb{Q}] + 1} } right)n · 4m. We show in fact that N(m) ≤ m + 4$nleftlceil {tfrac{m}n{4}} rightrceil n$nleftlceil {tfrac{m}n{4}} rightrceil n ([K: ℚ] − 1), our proof being constructive too. Moreover, we give some examples where this bound is sharp, for instance in the case of quadratic extensions. We also extend our results to the setting of non-commutative polynomials over ℚ.