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A set is called semidefinite representable or semidefinite programming (SDP)representable if it can be represented as the projection of a higherdimensional set which is represented by some Linear Matrix Inequality (LMI).This paper discuss the semidefinite representability conditions for convex setsof the form S_D(f) = {x \in D: f(x) >= 0}. Here D={x\in R^n: g_1(x) >= 0, ...,g_m(x) >= 0} is a convex domain defined by some "nice" concave polynomialsg_i(x) (they satisfy certain concavity certificates), and f(x) is a polynomialor rational function. When f(x) is concave over \mc{D}, we prove that S_D(f)has some explicit semidefinite representations under certain conditions calledpreordering concavity or q-module concavity, which are based on thePositivstellensatz certificates for the first order concavity criteria. Whenf(x) is a polynomial or rational function having singularities on the boundaryof S_D(f), a perspective transformation is introduced to find some explicitsemidefinite representations for S_D(f) under certain conditions. In theparticular case n=2, if the Laurent expansion of f(x) around one singular pointhas only two consecutive homogeneous parts, we show that S_D(f) always admitsan explicitly constructible semidefinite representation.

First Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular Polynomials