Sums of Squares over Totally Real Fields Are Rational Sums of Squares
نویسنده
چکیده
Let K be a totally real number field with Galois closure L. We prove that if f ∈ Q[x1, . . . , xn] is a sum of m squares in K[x1, . . . , xn], then f is a sum of 4m · 2[L:Q]+1 ([L : Q] + 1 2 ) squares in Q[x1, . . . , xn]. Moreover, our argument is constructive and generalizes to the case of commutative K-algebras. This result gives a partial resolution to a question of Sturmfels on the algebraic degree of certain semidefinite programing problems.
منابع مشابه
Bounding the Rational Sums of Squares over Totally Real Fields
Let K be a totally real Galois number field. C. J. Hillar proved that if f ∈ Q[x1, . . . , xn] is a sum of m squares in K[x1, . . . , xn], then f is a sum of N(m) squares in Q[x1, . . . , xn], where N(m) ≤ 2[K:Q]+1 · `[K:Q]+1 2 ́ ·4m, the proof being constructive. We show in fact that N(m) ≤ (4[K : Q]−3)·m, the proof being constructive as well.
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تاریخ انتشار 2008