ORIGINAL RESEARCH PAPER Continuous orthosymmetric multilinear maps and homogeneous polynomials on Riesz spaces Elmiloud Chil 1 • Abderraouf Dorai 2,3 Received: 4 July 2019 / Accepted: 31 March 2020 Ó Forum D’Analystes, Chennai 2020 Abstract We show that any continuous orthosymmetric multilinear map from an Archime- dean Riesz space into a Hausdorff topological vector space is symmetric. Then, we establish a linear representation for continuous orthogonally additive homogeneous polynomials. This representation will be used to introduce and describe a new class of homogeneous polynomials, namely that of polyorthomorphisms. In particular, we prove that, for a Riesz space E and a natural number n  2, the space P orth ð n EÞ of all polyorthomorphisms of degree n is a Riesz space. Keywords Riesz spaces  Relatively uniformly complete  Orthosymmetric multilinear maps  Homogeneous polynomials  Orthomorphisms  Polyorthomorphisms Mathematics Subject Classification 06F25  46A40 1 Introduction In recent years, there has been an increasing interest in orthosymmetric bilinear maps on Riesz spaces. Their importance derives from a similarity with Hilbert spaces theory, as well as the principal fact that they are symmetric. The notion of & Elmiloud Chil Elmiloud.chil@ipeit.rnu.tn Abderraouf Dorai abderraouf.dorai@ipeiem.utm.tn 1 Institut pre ´paratoire aux e ´tudes d’inge ´nieurs de Tunis, 2 Rue Jawaher lel Nahrou Montfleury, 1008 Tunis, Tunisia 2 L.A.T.A.O, Faculty of Sciences of Tunis, Tunis-El Manar University, Tunis, Tunisia 3 Abderraouf Dorai Institut pre ´paratoire aux e ´tudes d’inge ´nieurs El Manar, BP 244 El Manar, 2092 Tunis, Tunisia 123 The Journal of Analysis https://doi.org/10.1007/s41478-020-00240-2