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Topological and combinatoric methods for studying sums of squares Yiu, Paul Yu-Hung


We study sums of squares formulae from the perspective of normed bilinear maps and their Hopf constructions. We begin with the geometric properties of quadratic forms between euclidean spheres. Let F: Sm → Sn be a quadratic form. For every point q in the image, the inverse image F⁻¹ (q) is the intersection of Sm with a linear subspace wq, whose dimension can be determined easily. In fact, for every k ≤ m+1 with nonempty Yk = {q ∈ Sn: dim Wq = k}, the restriction F⁻¹ (Yk) → Yk is a great (k-1) - sphere bundle. The quadratic form F is the Hopf construction of a normed bilinear map if and only if it admits a pair of "poles" ±p such that dim Wp + dim W₋p = m+1. In this case, the inverse images of points on a "meridian", save possibly the poles, are mutually isoclinic. Furthermore, the collection of all poles forms a great sphere of relatively low dimension. We also prove that the classical Hopf fibrations are the only nonconstant quadratic forms which are harmonic morphisms in the sense that the composite with every real valued harmonic function is again harmonic. Hidden in a quadratic form F: Sm → Sn are nonsingular bilinear maps Rk x Rm-k⁺¹ → Rn, one for each point in the image, all representing the homotopy class of F, which lies in Im J. Moreover, every hidden nonsingular bilinear map can be homotoped to a normed bilinear map. The existence of one sums of squares formula, therefore, anticipates others which cannot be obtained simply by setting some of the indeterminates to zero. These geometric and topological properties of quadratic forms are then used, together with homotopy theory results in the literature, to deduce that certain sums of squares formulae cannot exist, notably of types [12,12,20] and [16,16,24]. We also prove that there is no nonconstant quadratic form S²⁵ → S²³. Sums of squares formulae with integer coefficients are equivalent to "intercalate matrices of colors with appropriate signs". This combinatorial nature enables us to establish a stronger nonexistence result: no sums of squares formula of type [16,16, 28] can exist if only integer coefficients are permitted. We also classify integral [10,10,16] formulae, and show that they all represent ±2Ʋ∈ π [s over 3]. With the aid of the KO theory of real projective spaces, we determine, for given δ ≤ 5 and s, the greatest possible r for which there exists an [r,s,s+δ] formula. An explicit solution of the classical Hurwitz-Radon matrix equations is also recorded.

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