TY - GEN A1 - Henk, Martin A1 - Averkov, Gennadiy T1 - Three-Dimensional Polyhedra Can Be Described by Three Polynomial Inequalities T2 - Discrete & computational geometry N2 - Bosse et al. conjectured that for every natural number d≥2 and every d-dimensional polytope P in ℝ d , there exist d polynomials p 1(x),…,p d (x) satisfying P={x∈ℝ d :p 1(x)≥0,…,p d (x)≥0}. We show that every three-dimensional polyhedron can be described by three polynomial inequalities, which confirms the conjecture for the case d=3 but also provides an analogous statement for the case of unbounded polyhedra. The proof of our result is constructive. KW - Łojasiewicz’s inequality KW - Polynomial KW - Polytope KW - Semi-algebraic set KW - Theorem of Bröcker and Scheiderer Y1 - 2009 UR - https://link.springer.com/article/10.1007%2Fs00454-009-9183-1 U6 - https://doi.org/10.1007/s00454-009-9183-1 SN - 1432-0444 VL - 42 IS - 2 SP - 166 EP - 186 ER - TY - GEN A1 - Averkov, Gennadiy A1 - Bey, Christian T1 - Description of polygonal regions by polynomials of bounded degree T2 - Monatshefte für Mathematik N2 - We show that every (possibly unbounded) convex polygon P in R2 with m edges can be represented by inequalities p 1 ≥ 0, . . ., p n ≥ 0, where the p i ’s are products of at most k affine functions each vanishing on an edge of P and n = n(m, k) satisfies s(m,k)≤n(m,k)≤(1+εm)s(m,k) with s(m,k) ≔ max {m/k, log2 m} and εm→0 as m→∞. This choice of n is asymptotically best possible. An analogous result on representing the interior of P in the form p 1 > 0, . . ., p n > 0 is also given. For k ≤ m/log2 m these statements remain valid for representations with arbitrary polynomials of degree not exceeding k. KW - Gray code KW - Polygon KW - Polynomial KW - Semi-algebraic set KW - Theorem of Bröcker and Scheiderer Y1 - 2011 U6 - https://doi.org/10.1007/s00605-010-0224-x SN - 1436-5081 VL - 162 IS - 1 SP - 19 EP - 27 ER -