Improved effective Łojasiewicz inequality and applications

Let  R  be a real closed field. Given a closed and bounded semialgebraic set  A⊂Rn  and semialgebraic continuous functions  f,g:A→R  such that  f−1(0)⊂g−1(0) , there exist an integer  N>0  and  c∈R  such that the inequality (Łojasiewicz inequality)  |g(x)|N≤c⋅|f(x)|  holds for all  x∈A . In this paper, we consider the case when A is defined by a quantifier-free formula with atoms of the form  P=0,P>0,P∈P  for some finite subset of polynomials  P⊂R[X1,…,Xn]≤d , and the graphs of  f,g  are also defined by quantifier-free formulas with atoms of the form  Q=0,Q>0,Q∈Q , for some finite set  Q⊂R[X1,…,Xn,Y]≤d . We prove that the Łojasiewicz exponent in this case is bounded by  (8d)2(n+7) . Our bound depends on d and n but is independent of the combinatorial parameters, namely the cardinalities of  P  and  Q . The previous best-known upper bound in this generality appeared in P. Solernó, Effective Łojasiewicz Inequalities in Semi-Algebraic Geometry, Applicable Algebra in Engineering, Communication and Computing (1991) and depended on the sum of degrees of the polynomials defining  A,f,g  and thus implicitly on the cardinalities of  P  and  Q . As a consequence, we improve the current best error bounds for polynomial systems under some conditions. Finally, we prove a version of Łojasiewicz inequality in polynomially bounded o-minimal structures. We prove the existence of a common upper bound on the Łojasiewicz exponent for certain combinatorially defined infinite (but not necessarily definable) families of pairs of functions. This improves a prior result of Chris Miller (C. Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic (1994)).