Expected linear time MST algorithm: Difference between revisions

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In [[mathematics]], the '''Carlitz exponential''' is a characteristic ''p'' analogue to the usual [[exponential function]] studied in [[real analysis|real]] and [[complex analysis]]. It is used in the definition of the [[Carlitz module]] – an example of a [[Drinfeld module]].
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==Definition==
 
We work over the polynomial ring '''F'''<sub>''q''</sub>[''T''] of one variable over a [[finite field]] '''F'''<sub>''q''</sub> with ''q'' elements. The [[Completion (metric space)|completion]] '''C'''<sub>∞</sub> of an [[algebraic closure]] of the field '''F'''<sub>''q''</sub>((''T''<sup>&minus;1</sup>)) of [[formal Laurent series]] in ''T''<sup>&minus;1</sup> will be useful. It is a complete and algebraically closed field.
 
First we need analogues to the [[factorials]], which appear in the definition of the usual exponential function. For ''i''&nbsp;>&nbsp;0 we define
 
:<math>[i] := T^{q^i} - T, \, </math>
:<math>D_i := \prod_{1 \le j \le i} [j]^{q^{i - j}}</math>
 
and ''D''<sub>0</sub> := 1. Note that that the usual factorial is inappropriate here, since ''n''! vanishes in '''F'''<sub>''q''</sub>[''T''] unless ''n'' is smaller than the [[Characteristic (algebra)|characteristic]] of '''F'''<sub>''q''</sub>[''T''].
 
Using this we define the Carlitz exponential ''e''<sub>''C''</sub>:'''C'''<sub>∞</sub>&nbsp;→&nbsp;'''C'''<sub>∞</sub> by the convergent sum
 
:<math>e_C(x) := \sum_{j = 0}^\infty \frac{x^{q^j}}{D_i}.</math>
 
==Relation to the Carlitz module==
 
The Carlitz exponential satisfies the functional equation
 
:<math>e_C(Tx) = Te_C(x) + \left(e_C(x)\right)^q = (T + \tau)e_C(x), \, </math>
 
where we may view <math> \tau </math> as the power of <math> q </math> map or as an element of the ring <math> F_q(T)\{\tau\} </math> of [[noncommutative polynomials]]. By the [[universal property]] of polynomial rings in one variable this extends to a ring homomorphism ''ψ'':'''F'''<sub>''q''</sub>[''T'']→'''C'''<sub>∞</sub>{''τ''}, defining a Drinfeld '''F'''<sub>''q''</sub>[''T'']-module over '''C'''<sub>∞</sub>{''τ''}. It is called the Carlitz module.
 
==References==
{{reflist}}
*{{Citation | last1=Goss | first1=D. | authorlink = David Goss | title=Basic structures of function field arithmetic | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] | isbn=978-3-540-61087-8 | mr=1423131 | year=1996 | volume=35}}
*{{Citation | last1=Thakur | first1=Dinesh | first2=S. | title=Function field arithmetic | publisher=[[World Scientific Publishing]] | location=New  Jersey| isbn=981-238-839-7 | mr=2091265 | year=2004}}
 
[[Category:Algebraic number theory]]
[[Category:Finite fields]]

Latest revision as of 09:43, 9 January 2015

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