2025/06/19 by Philipp J. di Dio, di Dio, Philipp J.
Mathematics · #44A60 #47A57 #90C22 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Holomorphic and Operator Theory #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA) #Optimization and Control (math.OC) #Primary 11E25 #Secondary 13J30
paper · pdf · doi:10.48550/arxiv.2506.16321
openalex publication_date 2025/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We study linear operators T:ℝ[x1,…,xn]→ℝ[x1,…,xn], especially for the purpose to move sets S⊆ℝ[x1,…,xn] into cones C⊆ℝ[x1,…,xn]: TS⊆ C. We develop the theory of (semi-)groups of operators (etA)t∈ℝ on ℝ[x1,…,xn], which requires techniques from regular Fréchet Lie groups. We study the special case of making non-negative polynomials Pos(K)≤ 2d with K⊆ℝn and int K≠ ∅ into sums of squares: TPos(K)≤ 2d⊆ ∑ℝ[x1,…,xn]≤ d2. With N:=dimℝ[x1,…,xn]≤ 2d = \binomn+2dn, for T, a memory of at most 2N+1 is required. Matrix multiplications TM, MT, T-1M, and MT-1 of T with any M∈ℝN× N require at most 4N2+1 operations. Transformations Tv and T-1v of vectors v∈ℝN require at most 4N+1 operations. Calculating T-1 of T requires only one (!) operation.