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, 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