1$ we expect that ``almost all'' specializations of the family \eqref{genform} to extensions of $\F_p(t)$ have Galois group all of $A_n$ or $S_n$. Another simple specialization of \eqref{genform} is $g_{p,k}(x) = x^{k p} - x + t$ for $p$ a prime and $k$ a positive integer. Changing variables via $x = 1/y$ and $t = 1/s$, the equation becomes $y^{k p} - s y^{k p -1} + s$. This is an Eisenstein polynomial with conductor $c = c_t + c_w = (kp-1) + (kp-1) = 2 k p - 2$. Thus $\F_p(t)[x]/g_{p,k}(x)$ is the function field of a genus zero curve, a fact which is also clear by the single global observation that $F[x]/g_{p,k}(x) = \F_p(x)$, as $t = - x^{k p} - x$. If $k = p^j$ then one has $g_{p,p^j}(x) = h_j(x) + t$ with $h_j(x)$ the Artin-Schreier polynomial $h_1(x) = x^p-x$ composed with itself $j$ times. Thus the Galois group of $g_{p,p^j}(x)$ is solvable. In the complementary case, computations with Frobenius elements using Jordan's criterion \eqref{Jordan} suggest that the Galois group of $g_{p,k}(x)$ is all of $A_{pk}$ or $S_{pk}$ except in the cases $g_{3,4}(x) = x^{12} - x + t$ and $g_{2,12}(x) = x^{24} - x + t$. Here the Galois groups are known to be the Mathieu group $M_{11}$ in its degree 12 representation and the Mathieu group $M_{24}$, respectively \cite[Theorems~6.6 and 6.3]{Ab}. So the evidence is strong that for each $p$, the field $\F_p(t)[x]/g_{p,k}(x)$ is in $\Fields^{\rm big}_{\F_p(t),pk,\{\infty\}}$ for infinitely many $k$. Abhyankar has studied many similar genus zero families; typically the focus is extracting rare examples with small Galois groups from families with generic Galois group $A_n$ or $S_n$. The same heuristic which supports Conjecture~\ref{mainconj} in characteristic zero gives two reasons why the corresponding statement fails in positive characteristic. Again one can interpret $\frac{1}{2} \prod_{v \in S} \lambda_{F_v,n}$ as the expected total mass of fields in $\Fields^{\rm big}_{F,n,S}$. But in the function field case if $n \geq p$ then each of the factors $\lambda_{F_v,n}$ is itself infinite; this is the phenomenon behind the existence of the family $g_{k}(x)$ above. Even if one bounds ramification somehow so that each $\lambda_{F_v,n}$ is replaced by a finite number $\lambda^*_{F_v,n}$, the numbers $\frac{1}{2} \prod_{v \in S} \lambda^*_{F_v,n}$ can still increase due to the lack of Archimedean places. This is the phenomenon behind the existence of the family of polynomials $g_{p,k}(x)$. \begin{thebibliography}{4} \bibitem{Ab} S. Abhyankar, Mathieu group coverings and linear group coverings, in M. D. Fried et. al. eds., {\em Recent Developments in the Inverse Galois Problem}, Contemporary Mathematics 186, American Mathematical Society, 1995, 293--319. \bibitem{Am} S.\ Amano, Eisenstein equations of degree $p$ in a ${\mathfrak p}$-adic field. {\em J. Fac. Sci. Univ. Tokyo Sect. IA Math.} {\bf 18} (1971), 1--21. \bibitem{Bh4} M. Bhargava, The density of discriminants of quartic rings and fields. {\em Annals of Math.} (2) {\bf 162} (2005), no. 2, 1031-1063. \bibitem{Bh} M.\ Bhargava, %Address at AMS-MAA meeting, Atlanta, January 2005. Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants. Forthcoming. \bibitem{Dr} E.\ Driver, {\em A targeted Martinet search}, Thesis, Arizona State University, 2006. \bibitem{GKS} F.\ Garvan, D.\ Kim, and D.\ Stanton, {Cranks and $t$-cores,} {\em Invent. Math.} {\bf 101} (1990), 1--17. \bibitem{Go} F.\ Q.\ Gouv\^ea, {\em $p$-adic Numbers: An Introduction}, Second edition, Springer, 1997. \bibitem{HR} G.\ H.\ Hardy and S.\ Ramanujan, Asymptotic formulae in combinatory analysis, {\em Proc.\ London Math.\ Soc.} {\bf 17} (1918), 75--115. \bibitem{Jo} C.\ Jordan, Sur la limite de transitivit\'{e} des groupes non-altern\'ees, {\em Bull.\ Soc.\ Math.\ France} {\bf 1} (1873), 40--71. \bibitem{JR6} J.\ W.\ Jones and D.\ P.\ Roberts, Sextic number fields with discriminant $-\sp j2\sp a3\sp b$, in R. Gupta and K. S. Williams, eds., {\em Number theory}, American Mathematical Society, 1999, 141--172. \bibitem{JR7} J.\ W.\ Jones and D.\ P.\ Roberts, Septic fields with discriminant $\pm 2\sp a3\sp b$, {\em Math. Comp.} {\bf 72} (2003), 1975--1985. \bibitem{JR} J.\ W.\ Jones and D. P.\ Roberts, A database of local fields, {\em J. Symbolic Comput.} {\bf 41} (2006), 80--97. Database at \href{http://math.la.asu.edu/~jj/localfields/}{http://math.la.asu.edu/$\sim$jj/localfields/} \bibitem{Ke} K. S. Kedlaya, Mass formulas for local Galois representations (with an appendix by Daniel Gulotta), {\em International Mathematics Research Notices}, {\bf 2007}, 25 pages. \bibitem{Ko} H.\ Koch, {\em Galois theory of $p$-extensions}, Springer, 2002. \bibitem{Kr} M.\ Krasner, Remarques au sujet d'une note de J.-P. Serre: Une ``formule de masse" pour les extensions totalement ramifi\'ees de degr\'{e} donn\'{e} d'un corps local, {\em C. R. Acad. Sci. Paris A-B} {\bf 288}, A863--A865. \bibitem{MR} G.\ Malle and D.\ P.\ Roberts, Number fields with discriminant $\pm 2\sp a3\sp b$ and Galois group $A\sb n$ or $S\sb n$, {\em LMS J. Comput. Math.} {\bf 8} (2005), 80--101. \bibitem{MW} L.\ Moser and M.\ Wyman, On solutions of $x^d=1$ in symmetric groups, {\em Canad.\ J.\ Math.} {\bf 7}, (1955), 159--168. \bibitem{NSW} J.\ Neukirch, A.\ Schmidt, and K.\ Wingberg, {\em Cohomology of number fields}, Springer, 2000. %\bibitem{PR} S. Pauli and X.-F.\ Roblot, On the computation of all extensions of a $p$-adic field of a given degree, {\em Math. Comp.} {\bf 70}, 1641--1659. \bibitem{Cubics} D. P.\ Roberts, Density of cubic field discriminants, {\em Math. Comp.} {\bf 70} (2001), 1699--1705. \bibitem{Ro} D.\ P.\ Roberts, An $ABC$ construction of number fields, in H. Kisilevsky and E. Z. Goren, eds., {\em Number theory}, CRM Proc. Lecture Notes, {\bf 36}, AMS, 2004, 237--267. \bibitem{Ro2} D.P.\ Roberts, Chebyshev covers and exceptional number fields. In preparation. \bibitem{SeCL} J.-P.\ Serre, {\em Local Fields}, Springer Graduate Texts in Mathematics 67, 1979. \bibitem{Se} J.-P.\ Serre, Une ``formule de masse" pour les extensions totalement ramifi\'ees de degr\'e donn\'e d'un corps local, {\em C. R. Acad. Sci. Paris A-B} {\bf 286} (1978), A1031--A1036. \bibitem{Sl} N.\ J.\ A. Sloane, {\em The On-Line Encyclopedia of Integer Sequences} at \\ \href{http://www.research.att.com/~njas/sequences/}{http://www.research.att.com/$\sim$njas/sequences/} \bibitem{St} R.\ P.\ Stanley, {\em Enumerative Combinatorics}, vol. 2, Cambridge, 1999. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 11S15; Secondary 11P72, 11R21.\\ \noindent {\it Keywords}: wild, partition, $p$-adic, ramified, mass. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A000041}, \seqnum{A000085}, \seqnum{A010054},\seqnum{A033687}, \seqnum{A131139}, and \seqnum{A131140}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received March 28 2007; revised version received June 18 2007. Published in {\it Journal of Integer Sequences}, June 18 2007. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}