Liouville-Łojasiewicz inequality
A Liouville inequality is one embodying the principle in number theory that algebraic numbers cannot be very well approximated by rational numbers or, equivalently, that integral polynomials cannot be small and non-zero at algebraic numbers (cf. also Liouville theorems). A Łojasiewicz inequality gives a lower bound for functions in terms of the distance to common zeros.
These features can be combined [a5] in the following Liouville–Łojasiewicz inequality. Let each 
have total degree at most 
and coefficients of absolute value at most 
. For 
, let 
be greater than or equal to the largest absolute value of the coordinates of 
and let 
be less than or equal to the distance from 
to the common zeros 
of 
. Then there are explicit constants 
, 
, 
depending on 
such that
 |
where 
, 
.
Over arbitrary fields with an absolute value, the lower bound takes the form 
, cf. [a4], [a2] and [a1] (in the last citation, the polynomials 
are replaced by ideals 
and 
are taken to be the values of fixed Chow coordinates of 
). In this setting, M. Hickel [a3] obtains the optimal involvement of 
at the right-hand side. Actually, the above arithmetic inequality holds with 
.
If, when working over 
, 
denotes a zero of an unmixed ideal 
and 
denotes the distance from 
to the zeros of 
, then the above upper bound holds with 
, 
, with 
replaced by 
, and 
by 
. When 
, the zeros 
of 
have algebraic coordinates. When 
and 
does not vanish at any point of 
, then one obtains an explicit lower bound on 
, i.e. a Liouville inequality.
References
| [a1] | J. Kollár, "Effective Nullstellensatz for arbitrary ideals" J. Europ. Math. Soc. (JEMS) , 1 (1999) pp. 313–337 |
| [a2] | S. Ji, J. Kollár, B. Shiffman, "A global Łojasiewicz inequality for algebraic varieties" Trans. Amer. Math. Soc. , 329 (1992) pp. 813–818 |
| [a3] | M. Hickel, "Solution d'une conjecture de C. Berenstein–A. Yger et invariants de contact à l'infini" Prepubl. Lab. Math. Pures Univ. Bordeaux I , 118 : jan. (2000) |
| [a4] | W.D. Brownawell, "Bounds for the degrees in the Nullstellensatz" Ann. of Math. , 126 (1987) pp. 577–591 |
| [a5] | W.D. Brownawell, "Local diophantine Nullstellen equalities" J. Amer. Math. Soc. , 1 (1988) pp. 311–322 |
Liouville-Łojasiewicz inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Liouville-%C5%81ojasiewicz_inequality&oldid=22761