Loyasevich Inequality

Loyasevich 's inequality is an inequality established by the Polish mathematician Stanislov Loyasevich ( Polish Stanisław Łojasiewicz ), giving an upper bound for the distance from the point of an arbitrary compact set to the zero-level set of a real analytic function of many variables. This inequality has found application in various branches of mathematics, including in real algebraic geometry, in analysis, in the theory of differential equations ^[1] ^[2] .

Wording

Let the function $f:U\to \mathbb {R}$ ${\ displaystyle f: U \ to \ mathbb {R}}$ is real analytic on a nonempty open set $U\subset \mathbb {R} ^{n}$ ${\ displaystyle U \ subset \ mathbb {R} ^ {n}}$ let it go $Z=\{x\in U:f(x)=0\}$ ${\ displaystyle Z = \ {x \ in U: f (x) = 0 \}}$ - set of zeros of the function $f$ ${\ displaystyle f}$ . If the set $Z$ ${\ displaystyle Z}$ nonempty, then for any nonempty compact $K\subset U$ ${\ displaystyle K \ subset U}$ there are such constants $\alpha \geq 2$ ${\ displaystyle \ alpha \ geq 2}$ and $C>0$ ${\ displaystyle C> 0}$ that inequality holds

\inf _{z\in Z}|x-z|^{\alpha }\leq C|f(x)|\ \ \forall \,x\in K,

{\ displaystyle \ inf _ {z \ in Z} | xz | ^ {\ alpha} \ leq C | f (x) | \ \ \ forall \, x \ in K,}

number $\alpha$ ${\ displaystyle \ alpha}$ which can be big enough.

Also for any point $p\in U$ ${\ displaystyle p \ in U}$ there is a sufficiently small neighborhood $W\subset U$ ${\ displaystyle W \ subset U}$ and such constants $0<\beta <1$ ${\ displaystyle 0 <\ beta <1}$ and $C>0$ ${\ displaystyle C> 0}$ that the second Loyasevich inequality holdsː

|f(x)-f(p)|^{\beta }\leq C|\nabla f(x)|\ \ \forall \,x\in W.

{\ displaystyle | f (x) -f (p) | ^ {\ beta} \ leq C | \ nabla f (x) | \ \ \ forall \, x \ in W.}

From the second inequality it obviously follows that for each critical point of a real analytic function there exists a neighborhood such that the function takes the same value at all critical points from this neighborhood.

Literature

Tobias Holck Colding, William P. Minicozzi II , Lojasiewicz inequalities and applications, arXiv: 1402.5087
Malgrange B. Ideals of differentiable functions. - M .: Mir, 1968.
Bierstone, Edward & Milman, Pierre D. (1988), " Semianalytic and subanalytic sets ", Publications Mathématiques de l'IHÉS (no. 67): 5–42, MR : 972342 , ISSN 1618-1913 , < http: // www.numdam.org/item?id=PMIHES_1988__67__5_0 >
Ji, Shanyu; Kollár, János & Shiffman, Bernard (1992), " A global Łojasiewicz inequality for algebraic varieties ", Transactions of the American Mathematical Society T. 329 (2): 813–818, MR : 1046016 , ISSN 0002-9947 , doi : 10.2307 / 2153965 , < http://www.ams.org/journals/tran/1992-329-02/S0002-9947-1992-1046016-6/ >

Notes

[1] V.I. Arnold, Yu.S. Ilyashenko . Ordinary differential equations, Dynamical systems - 1, Itogi Nauki i Tekhniki. Ser. Lying. prob. mat. Fundam. directions, 1, VINITI, M., 1985 .

[2] Yu. S. Ilyashenko, S. Yu. Yakovenko , Finite-smooth normal forms of local families of diffeomorphisms and vector fields, UMN, 46: 1 (277) (1991), 3–39 .

Loyasevich Inequality

Wording

Literature

Notes

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