View source for 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|Liouville theorems]]). A Łojasiewicz inequality gives a lower bound for functions in terms of the distance to common zeros.

These features can be combined [[#References|[a5]]] in the following Liouville–Łojasiewicz inequality. Let each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l1300801.png" /> have total degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l1300802.png" /> and coefficients of absolute value at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l1300803.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l1300804.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l1300805.png" /> be greater than or equal to the largest absolute value of the coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l1300806.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l1300807.png" /> be less than or equal to the distance from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l1300808.png" /> to the common zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l1300809.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008010.png" />. Then there are explicit constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008013.png" /> depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008014.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008015.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008017.png" />.

Over arbitrary fields with an absolute value, the lower bound takes the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008018.png" />, cf. [[#References|[a4]]], [[#References|[a2]]] and [[#References|[a1]]] (in the last citation, the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008019.png" /> are replaced by ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008021.png" /> are taken to be the values of fixed Chow coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008022.png" />). In this setting, M. Hickel [[#References|[a3]]] obtains the optimal involvement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008023.png" /> at the right-hand side. Actually, the above arithmetic inequality holds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008024.png" />.

If, when working over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008026.png" /> denotes a zero of an unmixed ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008028.png" /> denotes the distance from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008029.png" /> to the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008030.png" />, then the above upper bound holds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008032.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008033.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008034.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008035.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008036.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008037.png" />, the zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008039.png" /> have algebraic coordinates. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008041.png" /> does not vanish at any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008042.png" />, then one obtains an explicit lower bound on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130080/l13008043.png" />, i.e. a Liouville inequality.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Kollár,   "Effective Nullstellensatz for arbitrary ideals"  ''J. Europ. Math. Soc. (JEMS)'' , '''1'''  (1999)  pp. 313–337</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Ji,   J. Kollár,   B. Shiffman,   "A global Łojasiewicz inequality for algebraic varieties"  ''Trans. Amer. Math. Soc.'' , '''329'''  (1992)  pp. 813–818</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W.D. Brownawell,   "Bounds for the degrees in the Nullstellensatz"  ''Ann. of Math.'' , '''126'''  (1987)  pp. 577–591</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W.D. Brownawell,   "Local diophantine Nullstellen equalities"  ''J. Amer. Math. Soc.'' , '''1'''  (1988)  pp. 311–322</TD></TR></table>