On {\L}ojasiewicz Inequalities and the Effective Putinar's Positivstellensatz

The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semi-algebraic set $S$ and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter $\epsilon$ measuring the non-vanishing of the positive function, the constant $\mathfrak{c}$ and exponent $L$ of a {\L}ojasiewicz inequality for the semi-algebraic distance function associated to the inequalities $\mathbf{g} = (g_1, \dots , g_r)$ defining $S$. They are polynomial in $\mathfrak{c}$ and $\epsilon^{-1}$ with an exponent depending only on $L$. We analyse in details the {\L}ojasiewicz inequality when the defining inequalities $\mathbf g$ satisfy the Constraint Qualification Condition. We show that, in this case, the {\L}ojasiewicz exponent $L$ is $1$ and we relate the {\L}ojasiewicz constant $\mathfrak{c}$ with the distance of $\mathbf g$ to the set of singular systems.