@misc{HenkAverkov, author = {Henk, Martin and Averkov, Gennadiy}, title = {Three-Dimensional Polyhedra Can Be Described by Three Polynomial Inequalities}, series = {Discrete \& computational geometry}, volume = {42}, journal = {Discrete \& computational geometry}, number = {2}, issn = {1432-0444}, doi = {10.1007/s00454-009-9183-1}, pages = {166 -- 186}, abstract = {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.}, language = {en} } @misc{AverkovHenk, author = {Averkov, Gennadiy and Henk, Martin}, title = {Representing simple d-dimensional polytopes by d polynomials}, series = {Mathematical programming / A}, volume = {126}, journal = {Mathematical programming / A}, number = {2}, issn = {1436-4646}, doi = {10.1007/s10107-009-0280-y}, pages = {203 -- 230}, abstract = {A polynomial representation of a convex d-polytope P is a finite set {p 1(x), . . . , p n (x)} of polynomials over Rd such that P={x∈Rd:pi(x)≥0 for every 1≤i≤n}. Let s(d, P) be the least possible n as above. It is conjectured that s(d, P) = d for all convex d-polytopes P. We confirm this conjecture for simple d-polytopes by providing an explicit construction of d polynomials that represent a given simple d-polytope P.}, language = {en} }