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              In this quite long and very well structured article the authors introduce the notion of combinatorial hypersurfaces, which are codimension 2 submanifolds of $\\bbfC \\bbfP^n$ invariant under complex conjugation whose real parts are codimension 1 submanifolds of $\\bbfR \\bbfP^n$. This concept appears after removing the convexity condition imposed by Viro to the lattice subdivisions of the Newton polytope, to construct real algebraic varieties with prescribed topology [see e.g. {\\it O. Ya. Viro}, Russ. Math. Surv. 41, No. 3, 55-82 (1986)].\\par The authors show that combinatorial hypersurfaces obey almost all known topological restrictions satisfied by real algebraic surfaces; among them let us quote that they satisfy the generalized Harnack inequality, the Gudkov-Rokhlin and the Gudkov-Krahnov-Kharlamov congruences, some kind of Comessati inequalities for combinatorial hypersurfaces in $\\bbfC \\bbfP^3$, and that those of degree $d$ in $\\bbfC \\bbfP^3$ are homeomorphic to nonsingular algebraic surfaces in $\\bbfC \\bbfP^3$ of the same degree.\\par The paper can be viewed as the first step trying to answer the following questions:\\par (i) How far are are combinatorial hypersurfaces from the algebraic ones?\\par (ii) What are the main differences between the combinatorial hypersurfaces and the notion of flexible curve introduced by {\\it O. Ya. Viro} [in: Topology, general and algebraic topology, and applications. Proc. Int. Conf.,Leningrad 1982, Lect. Notes Math. 1060, 187-200 (1984)]?\\par It must be pointed out that, as the authors recognize, the notion of combinatorial hypersurface was firstly introduced, with an slightly different language, in the pioneer work of {\\it F. Santos} [``Improved counterexamples to the Ragsdale conjecture'', Univ. de Cantabria, Spain, Preprint 1994].
            """ ;
    dct:identifier "hal-00096965" ;
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    dct:publisher <https://rec.harvest-normandie.data4citizen.com/organization/cce9db95-46d9-4dc2-84b6-764215d0a002> ;
    dct:title "Viro theorem and topology of real and complex combinatorial hypersurfaces" ;
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    dcat:keyword "14p25-14j70-32c18",
        "combinatorial-hypersurfaces",
        "harnack-inequality",
        "infoeu-reposemanticsarticle",
        "journal-articles",
        "mathmath-gtmathematics-mathgeometric-topology-mathgt",
        "non-convex-subdivisions-of-newton-polytope" ;
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    dct:issued "2026-05-05T18:43:37.351647"^^xsd:dateTime ;
    dct:modified "2026-05-05T18:43:37.318190"^^xsd:dateTime ;
    dct:title "Viro theorem and topology of real and complex combinatorial hypersurfaces" ;
    dcat:accessURL <https://hal.science/hal-00096965> .

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