Extensions of theorems of Rattray and Makeev (Q1947837)

Call a function \(f: S^{n-1}\times S^{n-1}\to\mathbb{R}\) odd if \(f(-x,y)=f(x,-y)=-f(x,y)\), and symmetric if \(f(x,y)=f(y,x)\). The set \({\mathcal R}^{\mathrm{orth}}_{\mathrm{odd}}\subset\mathbb{N}^3\) consists of all \((n,m,k)\in\mathbb{N}^3\) with the property that for any collection \(f_1, \ldots, f_m: S^{n-1}\times S^{n-1}\to\mathbb{R}\) of odd continuous functions there exists an orthonormal \(k\)-frame \(e_1, \ldots, e_k\in\mathbb{R}^n\) such that \(f_l(e_i, e_j)=0\) for \(l=1,\ldots m, 1\leq i<j\leq k\). The set \({\mathcal R}_{\mathrm{odd}}\subset\mathbb{N}^3\) is defined by dropping the condition that the \(k\)-frame has to be orthonormal. The set \({\mathcal R}^{\mathrm{orth}}_{\mathrm{odd},\mathrm{sym}}\subset\mathbb{N}^3\) is defined analogously by considering continuous functions \(f_1, \ldots, f_m\) which are odd and symmetric. It has been proved implicitely by \textit{B. A. Rattray} [Ann. Math. (2) 60, 502--512 (1954; Zbl 0056.41804)] that \((n,1,n)\in{\mathcal R}^{\mathrm{orth}}_{\mathrm{odd},\mathrm{sym}}\). Clearly one has \({\mathcal R}^{\mathrm{orth}}_{\mathrm{odd}}\subset {\mathcal R}_{\mathrm{odd}}\) and \({\mathcal R}^{\mathrm{orth}}_{\mathrm{odd}} \subset {\mathcal R}^{\mathrm{orth}}_{\mathrm{odd},\mathrm{sym}}\). The generalized Rattray problem consists of determining these sets. The main result of the paper contains sufficient conditions for \((n,m,k)\) to lie in one of these sets. These conditions are expressed in terms of the algebra \(F_2[t_1, \ldots, t_k]\). For instance, if \(\prod_{1\leq i<j\leq k}(t_i+t_j)^{2m}\not\in \langle t^n_1, \ldots, t^n_k\rangle\) then \((n, m, k)\in {\mathcal R}_{\mathrm{odd}}\). Proofs are based on the Borel cohomology of Stiefel manifolds. The paper also contains generalizations of results from \textit{V. V. Makeev} [J. Math. Sci., New York 140, No. 4, 551-557 (2007); translation from Zap. Nauchn. Semin. POMI 329, 92-106 (2005; Zbl 1151.52304)] about equipartitions of probability distributions by hyperplanes.