Cross-positive linear maps, positive polynomials and sums of squaresKlep, Igor (Avtor) Šivic, Klemen (Avtor) Zalar, Aljaž (Avtor) positive polynomialssum of squarespositive mapscompletely positive mapsone-parameter semigroupsconvex conesA $\ast$-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.20262025-10-17 09:32:45Članek v reviji175128UDK: 517.9ISSN pri članku: 0021-8693DOI: 10.1016/j.jalgebra.2025.09.018COBISS_ID: 253591043sl