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We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half‐open decompositions and methods for interlacing polynomials we provide an explicit formula for the    h  *    ‐polynomial in case of complete bipartite graphs. In particular, we show that the    h  *    ‐polynomial is      ‐positive and real‐rooted. This proves Gal s conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthening due to Nevo and Petersen [On      ‐vectors satisfying the Kruskal–Katona inequalities.  Discrete Comput. Geom.   45 (3) (2011), 503–521].

ARITHMETIC ASPECTS OF SYMMETRIC EDGE POLYTOPES