Let
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=8f0bc7ea82a970bfd09c55d99f04e746" title="Click to view the MathML source">P,Q∈Fub>qub>[X]∖{0} be two coprime polynomials over the finite field
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si124.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=b9264fbb0cd6ddc57c9e30dc465fe92f" title="Click to view the MathML source">Fub>qub> with
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si11.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=4d8dafdf022fb32645b67ac6b75697f2" title="Click to view the MathML source">degP>degQ. We represent each polynomial
w over
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si124.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=b9264fbb0cd6ddc57c9e30dc465fe92f" title="Click to view the MathML source">Fub>qub> by
using a rational
base ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si5.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=e18775548425ff80b18ef905fe9327bb" title="Click to view the MathML source">P/Q and
digits ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si6.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=3c0274cac2d9c567c3d57a4aed0d93ac" title="Click to view the MathML source">sub>iub>∈Fub>qub>[X] satisfying
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si644.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=f8313c89bc82ca7a6cb85f0c012a3c6e" title="Click to view the MathML source">degsub>iub><degP.
Digit expansions of this type are also defined for formal La
urent series over
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si124.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=b9264fbb0cd6ddc57c9e30dc465fe92f" title="Click to view the MathML source">Fub>qub>. We prove
uniq
ueness and a
utomatic properties of these expansions. Altho
ugh the
ω -lang
uage of the possible digit strings is not reg
ular, we are able to characterize the digit expansions of algebraic elements. In partic
ular, we give a version of Christol's Theorem by showing that the digit string of the digit expansion of a formal La
urent series is a
utomatic if and only if the series is algebraic over
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si193.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=ab21ee811c9f8af161d6673fc9a73037" title="Click to view the MathML source">Fub>qub>[X]. Finally, we st
udy relations between digit expansions of formal La
urent series and a finite fields version of Mahler's 3/2-problem.