Lojasiewicz inequality

The Łojasiewicz inequality (mostly in German-language literature: Lojasiewicz inequality ; after Stanisław Łojasiewicz ) is an inequality of mathematical analysis that is mainly used in real algebraic geometry .

It clearly states that for an analytical function the distance of a point from the set of zeros of the function can be estimated as a function of the function value at this point. However, this interpretation must be viewed with caution, because the constants occurring in the inequality depend on the function and, depending on the choice of function, there can of course be small function values ​​at a greater distance from the set of zeros. ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle f^{-1}(0)}$ ${\displaystyle f}$ ${\displaystyle f(x)}$ ${\displaystyle f}$ ${\displaystyle f}$

General formulation

Be compact and be analytical functions with (defined on an open environment of ) . Then there are constants so that for all the inequality ${\displaystyle K\subset \mathbb {R} ^{n}}$ ${\displaystyle f,g\colon K\to \mathbb {R} }$ ${\displaystyle K}$ ${\displaystyle f^{-1}(0)\subset g^{-1}(0)}$ ${\displaystyle c,\alpha >0}$ ${\displaystyle x\in K}$

${\displaystyle |f(x)|\geq c|g(x)|^{\alpha }}$

applies.

Distance to the number of zeros

The general formulation can particularly be applied to, because for this function is . The following corollary is obtained. ${\displaystyle g(x)=d(x,f^{-1}(0))}$ ${\displaystyle g^{-1}(0)=f^{-1}(0)}$

For every analytic function on an open neighborhood of a compact set there are constants , so that for all the inequality ${\displaystyle K\subset \mathbb {R} ^{n}}$ ${\displaystyle f\colon K\to \mathbb {R} }$ ${\displaystyle c,\alpha >0}$ ${\displaystyle x\in K}$

${\displaystyle |f(x)|\geq cd(x,f^{-1}(0))^{\alpha }}$

applies.

literature

• Edward Bierstone, Pierre Milman: Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math. No. 1988, 67: 5-42.