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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.
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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
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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, see also [[Lojasiewicz inequality]].
  
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" />.
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These features can be combined [[#References|[a5]]] in the following Liouville–Łojasiewicz inequality. Let each $P _ { 1 } , \ldots , P _ { m } \in \mathbf{Z} [ x _ { 1 } , \ldots , x _ { n } ]$ have total degree at most $d$ and coefficients of absolute value at most $\operatorname{exp} (  h )$. For $\omega \in \mathbf{C} ^ { n }$, let $| \omega | \geq 1$ be greater than or equal to the largest absolute value of the coordinates of $\omega$ and let $\rho \leq 1$ be less than or equal to the distance from $\omega$ to the common zeros $Z$ of $P_ i$. Then there are explicit constants $c_1$, $c_2$, $c_{3}$ depending on $n$ such that
  
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" />.
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\begin{equation*} \operatorname { log } \operatorname { max } \{ | P _ { i } ( \omega ) | \} \geq - d ^ { \mu } ( c _ { 1 } d + c _ { 2 } h ) + c _ { 3 } d ^ { \nu } \operatorname { log } \frac { \rho } { | \omega | }, \end{equation*}
  
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.
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where $\mu : = \operatorname { min } \{ m , n - 1 \}$, $\nu : = \operatorname { min } \{ m , n \}$.
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Over arbitrary fields with an absolute value, the lower bound takes the form $-  c  _ { 1 } +  c _ { 3 } d ^ { \nu } \operatorname { log } ( \rho / | \omega | )$, cf. [[#References|[a4]]], [[#References|[a2]]] and [[#References|[a1]]] (in the last citation, the polynomials $P_ i$ are replaced by ideals $I_i$ and $I _ { i } ( \omega )$ are taken to be the values of fixed Chow coordinates of $I_i$). In this setting, M. Hickel [[#References|[a3]]] obtains the optimal involvement of $| \omega |$ at the right-hand side. Actually, the above arithmetic inequality holds with $c _ { 3 } = 1$.
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If, when working over $\bf Z$, $\omega$ denotes a zero of an unmixed ideal $I$ and $\rho$ denotes the distance from $\omega$ to the zeros of $I + ( P _ { 1 } , \dots , P _ { m } )$, then the above upper bound holds with $\mu : = \operatorname { min } \{ \operatorname { dim } I , n - 1 \}$, $\nu : = \operatorname { min } \{ \operatorname { dim } I , n \}$, with $c_1$ replaced by $c_1 \operatorname{deg} I + c _ { 2 } \operatorname{log} \operatorname{ht} I$, and $c_2$ by $c - 2 \operatorname { deg } I$. When $\operatorname { dim } I = 0$, the zeros $Z ( l )$ of $I$ have algebraic coordinates. When $m = 1$ and $P _ { 1 }$ does not vanish at any point of $Z ( l )$, then one obtains an explicit lower bound on $| P _ { 1 } ( \omega ) |$, i.e. a Liouville inequality.
  
 
====References====
 
====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>
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<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>

Latest revision as of 16:56, 1 July 2020

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, see also Lojasiewicz inequality.

These features can be combined [a5] in the following Liouville–Łojasiewicz inequality. Let each $P _ { 1 } , \ldots , P _ { m } \in \mathbf{Z} [ x _ { 1 } , \ldots , x _ { n } ]$ have total degree at most $d$ and coefficients of absolute value at most $\operatorname{exp} ( h )$. For $\omega \in \mathbf{C} ^ { n }$, let $| \omega | \geq 1$ be greater than or equal to the largest absolute value of the coordinates of $\omega$ and let $\rho \leq 1$ be less than or equal to the distance from $\omega$ to the common zeros $Z$ of $P_ i$. Then there are explicit constants $c_1$, $c_2$, $c_{3}$ depending on $n$ such that

\begin{equation*} \operatorname { log } \operatorname { max } \{ | P _ { i } ( \omega ) | \} \geq - d ^ { \mu } ( c _ { 1 } d + c _ { 2 } h ) + c _ { 3 } d ^ { \nu } \operatorname { log } \frac { \rho } { | \omega | }, \end{equation*}

where $\mu : = \operatorname { min } \{ m , n - 1 \}$, $\nu : = \operatorname { min } \{ m , n \}$.

Over arbitrary fields with an absolute value, the lower bound takes the form $- c _ { 1 } + c _ { 3 } d ^ { \nu } \operatorname { log } ( \rho / | \omega | )$, cf. [a4], [a2] and [a1] (in the last citation, the polynomials $P_ i$ are replaced by ideals $I_i$ and $I _ { i } ( \omega )$ are taken to be the values of fixed Chow coordinates of $I_i$). In this setting, M. Hickel [a3] obtains the optimal involvement of $| \omega |$ at the right-hand side. Actually, the above arithmetic inequality holds with $c _ { 3 } = 1$.

If, when working over $\bf Z$, $\omega$ denotes a zero of an unmixed ideal $I$ and $\rho$ denotes the distance from $\omega$ to the zeros of $I + ( P _ { 1 } , \dots , P _ { m } )$, then the above upper bound holds with $\mu : = \operatorname { min } \{ \operatorname { dim } I , n - 1 \}$, $\nu : = \operatorname { min } \{ \operatorname { dim } I , n \}$, with $c_1$ replaced by $c_1 \operatorname{deg} I + c _ { 2 } \operatorname{log} \operatorname{ht} I$, and $c_2$ by $c - 2 \operatorname { deg } I$. When $\operatorname { dim } I = 0$, the zeros $Z ( l )$ of $I$ have algebraic coordinates. When $m = 1$ and $P _ { 1 }$ does not vanish at any point of $Z ( l )$, then one obtains an explicit lower bound on $| P _ { 1 } ( \omega ) |$, 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
How to Cite This Entry:
Liouville-Łojasiewicz inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Liouville-%C5%81ojasiewicz_inequality&oldid=22761
This article was adapted from an original article by W. Dale Brownawell (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article