20.500.12556/RUL-175128 Cross-positive linear maps, positive polynomials and sums of squares A $\ast$-linear map $\Phi$ between matrix spaces is cross-positive if it is positive on orthogonal pairs $(U,V)$ of positive semidefinite matrices in the sense that $\langle U,V \rangle:={\rm tr}(UV)=0$ implies $\langle\Phi (U),V \rangle \ge 0$, and is completely cross-positive if all its ampliations $I_n \otimes \Phi$ are cross-positive. (Completely) cross-positive maps arise in the theory of operator semigroups, where they are sometimes called exponentially-positive maps, and are also important in the theory of affine processes on symmetric cones in mathematical finance. To each $\Phi$ as above a bihomogeneous form is associated by $p_\Phi (x,y)=y^T\Phi (xx^T)y$. Then $\Phi$ is cross-positive if and only if $p_\Phi$ is nonnegative on the variety of pairs of orthogonal vectors $\{(x,y) | x^Ty = 0\}$. Moreover, $\Phi$ is shown to be completely cross-positive if and only if $p_\Phi$ is a sum of squares modulo the principal ideal $(x^Ty)$. These observations bring the study of cross-positive maps into the powerful setting of real algebraic geometry. Here this interplay is exploited to prove quantitative bounds on the fraction of cross-positive maps that are completely cross-positive. Detailed results about cross-positive maps $\Phi$ mapping between $3\times3$ matrices are given. Finally, an algorithm to produce cross-positive maps that are not completely cross-positive is presented. positive polynomials sum of squares positive maps completely positive maps one-parameter semigroups convex cones pozitivni polinomi vsote kvadratov pozitivne preslikave popolnoma pozitivne preslikave enoparametrične polgrupe konveksni stožci true false true Angleški jezik Slovenski jezik Članek v reviji 2025-10-17 09:32:45 2025-10-17 09:32:54 2026-07-01 14:27:20 0000-00-00 00:00:00 2026 0 0 Str. 189-243 Vol. 688 Feb. 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 517.9 0021-8693 10.1016/j.jalgebra.2025.09.018 253591043 1-s2.0-S0021869325005526-main.pdf 1-s2.0-S0021869325005526-main.pdf 1 9626DEDBCDD2F019134905524E62F349 15c9c2fc5f0ffde9bc3630a9b2994bccdd6738fb9c3f18f3b3f1808c6977d221 139d5bc4-ab2c-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=220413 https://www.sciencedirect.com/science/article/pii/S0021869325005526 1 3a41e03d-ab2b-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=220411 Fakulteta za matematiko in fiziko Fakulteta za računalništvo in informatiko 0 0 0