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We examine linear programming (LP) based relaxations for synthesizing polynomial Lyapunov functions to prove the stability of polynomial ordinary differential equations (ODEs). Our approach starts from a desired parametric polynomial form of the polynomial Lyapunov function. Subsequently, we encode the positive definiteness of the function, and the negation of its derivative, over the domain of interest. We first compare two classes of relaxations for encoding polynomial positivity: relaxations by sum-of-squares (SOS) programmes, against relaxations based on Handelman representations and Bernstein polynomials, that produce linear programmes. Next, we present a series of increasingly powerful LP relaxations based on expressing the given polynomial in its Bernstein form, as a linear combination of Bernstein polynomials. Subsequently, we show how these LP relaxations can be used to search for …