View source for Masser-Philippon/Lazard-Mora example

''Lazard–Mora/Masser–Philippon example, Lazard–Mora example, Masser–Philippon example''

An extremal family for the degrees in the Hilbert Nullstellensatz (cf. [[Hilbert theorem|Hilbert theorem]]) is given by the following example, ascribed variously to D.W. Masser and P. Philippon and to D. Lazard and T. Mora:

<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/m/m130/m130060/m1300601.png" /></td> </tr></table>

<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/m/m130/m130060/m1300602.png" /></td> </tr></table>

The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130060/m1300603.png" /> are readily seen to have no common zeros. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130060/m1300604.png" /> are polynomials 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/m/m130/m130060/m1300605.png" /></td> </tr></table>

by evaluation on the rational curve

<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/m/m130/m130060/m1300606.png" /></td> </tr></table>

it is easy to see that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130060/m1300607.png" />. This lower bound of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130060/m1300608.png" /> for the degrees of the coefficients for the Nullstellensatz is much better than the doubly exponential lower bound for the general ideal membership problem given in [[#References|[a2]]]. Variants of the example, cf. [[#References|[a3]]], show that the terms in the [[Liouville–Łojasiewicz inequality|Liouville–Łojasiewicz inequality]] are nearly optimal, with the presumed exception depending solely on the degree.

Another family of extremal examples for the Nullstellensatz is given in [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Kollár,   "Sharp effective Nullstellensatz"  ''J. Amer. Math. Soc.'' , '''1'''  (1988)  pp. 963–975</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.W. Mayr,   A.R. Meyer,   "The complexity of the word problems in commutative semigroups and polynlomial ideals"  ''Adv. Math.'' , '''46'''  (1982)  pp. 305–329</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.D. Brownawell,   "Local diophantine Nullstellen equalities"  ''J. Amer. Math. Soc.'' , '''1'''  (1988)  pp. 311–322</TD></TR></table>