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Effective lower bounds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o1300201.png" />, the minimal value of the [[Discriminant|discriminant]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o1300202.png" /> of algebraic number fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o1300203.png" /> having signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o1300204.png" /> (i.e. having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o1300205.png" /> real and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o1300206.png" /> non-real conjugates), obtained in 1976 by A.M. Odlyzko. See also [[Algebraic number|Algebraic number]]; [[Number field|Number field]].
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Effective lower bounds for $M ( r _ { 1 } , r _ { 2 } )$, the minimal value of the [[Discriminant|discriminant]] $| d ( K ) |$ of algebraic number fields $K$ having signature $( r _ { 1 } , r _ { 2 } )$ (i.e. having $r_1$ real and $2r_2$ non-real conjugates), obtained in 1976 by A.M. Odlyzko. See also [[Algebraic number|Algebraic number]]; [[Number field|Number field]].
  
 
The first such bound was proved in 1891 by H. Minkowski [[#References|[a4]]], who showed
 
The first such bound was proved in 1891 by H. Minkowski [[#References|[a4]]], who showed
  
<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/o/o130/o130020/o1300207.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} M ( r _ { 1 } , r _ { 2 } ) &gt; \left( \frac { \pi } { 4 } \right) ^ { 2 r _ { 2 } } \left( \frac { n ^ { n } } { n ! } \right) ^ { 2 }, \end{equation}
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o1300208.png" />. He obtained it using methods from the [[Geometry of numbers|geometry of numbers]]; the same method was used later by several authors to improve (a1) (see [[#References|[a5]]] for the strongest result obtained in this way).
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with $n = r _ { 1 } + 2 r _ { 2 }$. He obtained it using methods from the [[Geometry of numbers|geometry of numbers]]; the same method was used later by several authors to improve (a1) (see [[#References|[a5]]] for the strongest result obtained in this way).
  
In 1974, H.M. Stark ([[#References|[a11]]], [[#References|[a12]]]) observed that Hadamard factorization of the [[Dedekind zeta-function|Dedekind zeta-function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o1300209.png" /> leads to a formula expressing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002010.png" /> by the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002011.png" /> and the value of its logarithmic derivative at a complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002012.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002013.png" />. He used this formula with a proper choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002014.png" /> to deduce lower bounds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002015.png" /> which were essentially stronger than Minkowski's bound, but did not reach the bounds obtained by geometrical methods.
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In 1974, H.M. Stark ([[#References|[a11]]], [[#References|[a12]]]) observed that Hadamard factorization of the [[Dedekind zeta-function|Dedekind zeta-function]] $\zeta _ { K } ( s )$ leads to a formula expressing $\operatorname{log} | d ( K ) |$ by the zeros of $\zeta _ { K } ( s )$ and the value of its logarithmic derivative at a complex number $s _ { 0 } \neq 0,1$ with $\zeta_{ K } ( s _ { 0 } ) \neq 0$. He used this formula with a proper choice of $s_0$ to deduce lower bounds for $M ( r _ { 1 } , r _ { 2 } )$ which were essentially stronger than Minkowski's bound, but did not reach the bounds obtained by geometrical methods.
  
 
In 1976, Odlyzko [[#References|[a7]]] (cf. [[#References|[a9]]]) modified Stark's formula and obtained the following important improvement of (a1):
 
In 1976, Odlyzko [[#References|[a7]]] (cf. [[#References|[a9]]]) modified Stark's formula and obtained the following important improvement of (a1):
  
<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/o/o130/o130020/o13002016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002016.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a2)</td></tr></table>
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002017.png" />.
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with $\operatorname { lim } _ { x \rightarrow \infty } \epsilon ( n ) = 0$.
  
 
In particular, one has
 
In particular, one has
  
<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/o/o130/o130020/o13002018.png" /></td> </tr></table>
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\begin{equation*} D = \liminf  _ { n \rightarrow \infty } M ( r _ { 1 } , r _ { 2 } ) ^ { 1 / n } \geq 22. \end{equation*}
  
If the extended Riemann hypothesis is assumed (cf. also [[Riemann hypotheses|Riemann hypotheses]]; [[Zeta-function|Zeta-function]]), then the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002020.png" /> in (a2) can be replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002022.png" />, respectively. For small degrees the bound (a2) can be improved (see [[#References|[a2]]], [[#References|[a10]]]) and several exact values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002023.png" /> are known.
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If the extended Riemann hypothesis is assumed (cf. also [[Riemann hypotheses|Riemann hypotheses]]; [[Zeta-function|Zeta-function]]), then the constants $60$ and $22$ in (a2) can be replaced by $180$ and $41$, respectively. For small degrees the bound (a2) can be improved (see [[#References|[a2]]], [[#References|[a10]]]) and several exact values of $M ( r _ { 1 } , r _ { 2 } )$ are known.
  
On the other hand, it has been shown in [[#References|[a13]]], as a consequence of their solution of the class field tower problem (cf. also [[Tower of fields|Tower of fields]]; [[Class field theory|Class field theory]]), that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002024.png" /> is finite. The best explicit upper bound for it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002025.png" />, is due to J. Martinet [[#References|[a1]]], who obtained this as a corollary of his constructions of infinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002026.png" />-class towers of suitable fields.
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On the other hand, it has been shown in [[#References|[a13]]], as a consequence of their solution of the class field tower problem (cf. also [[Tower of fields|Tower of fields]]; [[Class field theory|Class field theory]]), that $D$ is finite. The best explicit upper bound for it, $D \leq 92.4$, is due to J. Martinet [[#References|[a1]]], who obtained this as a corollary of his constructions of infinite $2$-class towers of suitable fields.
  
 
For surveys of this topic, see [[#References|[a9]]], [[#References|[a3]]] and [[#References|[a8]]].
 
For surveys of this topic, see [[#References|[a9]]], [[#References|[a3]]] and [[#References|[a8]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Martinet,  "Tours de corps de classes et estimations de discriminants"  ''Invent. Math.'' , '''44'''  (1978)  pp. 65–73</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Martinet,  "Petits discriminants"  ''Ann. Inst. Fourier (Grenoble)'' , '''29''' :  fasc.1  (1979)  pp. 159–170</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Martinet,  "Petits discriminants des corps de nombres" , ''Journ. Arithm. 1980'' , Cambridge Univ. Press  (1982)  pp. 151–193</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Minkowski,  "Théorèmes arithmétiques"  ''C.R. Acad. Sci. Paris'' , '''112'''  (1891)  pp. 209–212  (Gesammelte Abh. I (1911), 261-263, Leipzig–Berlin)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H.P. Mulholland,  "On the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130020/o13002027.png" /> complex homogeneous linear forms"  ''J. London Math. Soc.'' , '''35'''  (1960)  pp. 241–250</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Odlyzko,  "Some analytic estimates of class numbers and discriminants"  ''Invent. Math.'' , '''29'''  (1975)  pp. 275–286</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Odlyzko,  "Lower bounds for discriminants of number fields"  ''Acta Arith.'' , '''29'''  (1976)  pp. 275–297  (II: Tôhoku Math. J., 29 (1977), 275-286)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Odlyzko,  "Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results"  ''Sém. de Théorie des Nombres, Bordeaux'' , '''2'''  (1990)  pp. 119–141</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  G. Poitou,  "Minoration de discriminants (d'aprés A.M. Odlyzko)" , ''Sém. Bourbaki (1975/76)'' , ''Lecture Notes in Mathematics'' , '''567''' , Springer  (1977)  pp. 136–153</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  G. Poitou,  "Sur les petits discriminants"  ''Sém. Delange–Pisot–Poitou'' , '''18''' :  6  (1976/77)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  H.M. Stark,  "Some effective cases of the Brauer–Siegel theorem"  ''Invent. Math.'' , '''23'''  (1974)  pp. 135–152</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  H.M. Stark,  "The analytic theory of numbers"  ''Bull. Amer. Math. Soc.'' , '''81'''  (1975)  pp. 961–972,</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  E.S. Golod,  I.R. Shafarevich,  "On the class-field tower"  ''Izv. Akad. Nauk. SSSR'' , '''28'''  (1964)  pp. 261–272  (In Russian)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  J. Martinet,  "Tours de corps de classes et estimations de discriminants"  ''Invent. Math.'' , '''44'''  (1978)  pp. 65–73</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Martinet,  "Petits discriminants"  ''Ann. Inst. Fourier (Grenoble)'' , '''29''' :  fasc.1  (1979)  pp. 159–170</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Martinet,  "Petits discriminants des corps de nombres" , ''Journ. Arithm. 1980'' , Cambridge Univ. Press  (1982)  pp. 151–193</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  H. Minkowski,  "Théorèmes arithmétiques"  ''C.R. Acad. Sci. Paris'' , '''112'''  (1891)  pp. 209–212  (Gesammelte Abh. I (1911), 261-263, Leipzig–Berlin)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  H.P. Mulholland,  "On the product of $n$ complex homogeneous linear forms"  ''J. London Math. Soc.'' , '''35'''  (1960)  pp. 241–250</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  A. Odlyzko,  "Some analytic estimates of class numbers and discriminants"  ''Invent. Math.'' , '''29'''  (1975)  pp. 275–286</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  A. Odlyzko,  "Lower bounds for discriminants of number fields"  ''Acta Arith.'' , '''29'''  (1976)  pp. 275–297  (II: Tôhoku Math. J., 29 (1977), 275-286)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  A. Odlyzko,  "Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results"  ''Sém. de Théorie des Nombres, Bordeaux'' , '''2'''  (1990)  pp. 119–141</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  G. Poitou,  "Minoration de discriminants (d'aprés A.M. Odlyzko)" , ''Sém. Bourbaki (1975/76)'' , ''Lecture Notes in Mathematics'' , '''567''' , Springer  (1977)  pp. 136–153</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  G. Poitou,  "Sur les petits discriminants"  ''Sém. Delange–Pisot–Poitou'' , '''18''' :  6  (1976/77)</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  H.M. Stark,  "Some effective cases of the Brauer–Siegel theorem"  ''Invent. Math.'' , '''23'''  (1974)  pp. 135–152</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  H.M. Stark,  "The analytic theory of numbers"  ''Bull. Amer. Math. Soc.'' , '''81'''  (1975)  pp. 961–972,</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  E.S. Golod,  I.R. Shafarevich,  "On the class-field tower"  ''Izv. Akad. Nauk. SSSR'' , '''28'''  (1964)  pp. 261–272  (In Russian)</td></tr></table>

Revision as of 16:59, 1 July 2020

Effective lower bounds for $M ( r _ { 1 } , r _ { 2 } )$, the minimal value of the discriminant $| d ( K ) |$ of algebraic number fields $K$ having signature $( r _ { 1 } , r _ { 2 } )$ (i.e. having $r_1$ real and $2r_2$ non-real conjugates), obtained in 1976 by A.M. Odlyzko. See also Algebraic number; Number field.

The first such bound was proved in 1891 by H. Minkowski [a4], who showed

\begin{equation} \tag{a1} M ( r _ { 1 } , r _ { 2 } ) > \left( \frac { \pi } { 4 } \right) ^ { 2 r _ { 2 } } \left( \frac { n ^ { n } } { n ! } \right) ^ { 2 }, \end{equation}

with $n = r _ { 1 } + 2 r _ { 2 }$. He obtained it using methods from the geometry of numbers; the same method was used later by several authors to improve (a1) (see [a5] for the strongest result obtained in this way).

In 1974, H.M. Stark ([a11], [a12]) observed that Hadamard factorization of the Dedekind zeta-function $\zeta _ { K } ( s )$ leads to a formula expressing $\operatorname{log} | d ( K ) |$ by the zeros of $\zeta _ { K } ( s )$ and the value of its logarithmic derivative at a complex number $s _ { 0 } \neq 0,1$ with $\zeta_{ K } ( s _ { 0 } ) \neq 0$. He used this formula with a proper choice of $s_0$ to deduce lower bounds for $M ( r _ { 1 } , r _ { 2 } )$ which were essentially stronger than Minkowski's bound, but did not reach the bounds obtained by geometrical methods.

In 1976, Odlyzko [a7] (cf. [a9]) modified Stark's formula and obtained the following important improvement of (a1):

(a2)

with $\operatorname { lim } _ { x \rightarrow \infty } \epsilon ( n ) = 0$.

In particular, one has

\begin{equation*} D = \liminf _ { n \rightarrow \infty } M ( r _ { 1 } , r _ { 2 } ) ^ { 1 / n } \geq 22. \end{equation*}

If the extended Riemann hypothesis is assumed (cf. also Riemann hypotheses; Zeta-function), then the constants $60$ and $22$ in (a2) can be replaced by $180$ and $41$, respectively. For small degrees the bound (a2) can be improved (see [a2], [a10]) and several exact values of $M ( r _ { 1 } , r _ { 2 } )$ are known.

On the other hand, it has been shown in [a13], as a consequence of their solution of the class field tower problem (cf. also Tower of fields; Class field theory), that $D$ is finite. The best explicit upper bound for it, $D \leq 92.4$, is due to J. Martinet [a1], who obtained this as a corollary of his constructions of infinite $2$-class towers of suitable fields.

For surveys of this topic, see [a9], [a3] and [a8].

References

[a1] J. Martinet, "Tours de corps de classes et estimations de discriminants" Invent. Math. , 44 (1978) pp. 65–73
[a2] J. Martinet, "Petits discriminants" Ann. Inst. Fourier (Grenoble) , 29 : fasc.1 (1979) pp. 159–170
[a3] J. Martinet, "Petits discriminants des corps de nombres" , Journ. Arithm. 1980 , Cambridge Univ. Press (1982) pp. 151–193
[a4] H. Minkowski, "Théorèmes arithmétiques" C.R. Acad. Sci. Paris , 112 (1891) pp. 209–212 (Gesammelte Abh. I (1911), 261-263, Leipzig–Berlin)
[a5] H.P. Mulholland, "On the product of $n$ complex homogeneous linear forms" J. London Math. Soc. , 35 (1960) pp. 241–250
[a6] A. Odlyzko, "Some analytic estimates of class numbers and discriminants" Invent. Math. , 29 (1975) pp. 275–286
[a7] A. Odlyzko, "Lower bounds for discriminants of number fields" Acta Arith. , 29 (1976) pp. 275–297 (II: Tôhoku Math. J., 29 (1977), 275-286)
[a8] A. Odlyzko, "Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results" Sém. de Théorie des Nombres, Bordeaux , 2 (1990) pp. 119–141
[a9] G. Poitou, "Minoration de discriminants (d'aprés A.M. Odlyzko)" , Sém. Bourbaki (1975/76) , Lecture Notes in Mathematics , 567 , Springer (1977) pp. 136–153
[a10] G. Poitou, "Sur les petits discriminants" Sém. Delange–Pisot–Poitou , 18 : 6 (1976/77)
[a11] H.M. Stark, "Some effective cases of the Brauer–Siegel theorem" Invent. Math. , 23 (1974) pp. 135–152
[a12] H.M. Stark, "The analytic theory of numbers" Bull. Amer. Math. Soc. , 81 (1975) pp. 961–972,
[a13] E.S. Golod, I.R. Shafarevich, "On the class-field tower" Izv. Akad. Nauk. SSSR , 28 (1964) pp. 261–272 (In Russian)
How to Cite This Entry:
Odlyzko bounds. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Odlyzko_bounds&oldid=15182
This article was adapted from an original article by Władysław Narkiewicz (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article