Document Type : Research Paper


Shiraz University


Let $K$ be a field of characteristic
$p>0$, $K[[x]]$, the ring of formal power series over $ K$,
$K((x))$, the quotient field of $ K[[x]]$, and $ K(x)$ the field
of rational functions over $K$. We shall give some
characterizations of an algebraic function $f\in K((x))$ over $K$.
Let $L$ be a field of characteristic zero. The power series $f\in
L[[x]]$ is called differentially algebraic, if it satisfies a
differential equation of the form $P(x, y, y',...)=0$, where $P$
is a non-trivial polynomial. This notion is defined over fields of
characteristic zero and is not so significant over fields of
characteristic $p>0$, since $f^{(p)}=0$. We shall define an
analogue of the concept of a differentially algebraic power series
over $K$ and we shall find some more related results.


[1] G. Christol, T. Kamae, M. Mendes-France, and G. Rauzy, Suites algebriques, automates et
substitutions, Bull. Soc. Math. France 108 (1980)401-419.
[2]J. Denef and L. Lipshitz, Algebraic power series and diagonals, J. Number Theory 26 (1987)
[3] H. Furstenberg, Algebraic functions over finite fields, J. Algebra 7 (1967)271-277.
[4] N. Koblitz, p-adic analysis; a short course on recent work, Cambridge U. P.; LMS Lecture Notes
Series 46, 1980.
[5] L. Lipshitz, The diagonal of a D-finite power series is D-finite, J. Algebra 113 (1988)373-378.
[6] L. Lipshitz and L. Rubel, A gap theorem for power series solutions of algebraic differential
equations, Amer. J. Math., 108 (1986) 1193-1214.
[7] M. Mendes-France and A. J. van der Poorten, Automata and the arithmetic of formal power
series, Acta Arith 46 (1986)211-214.
[8] H. Sharif, Algebraic functions, differentially algebraic power series and Hadamard operations,
Ph.D. Thesis, Kent, 1989.
[9] —, Algebraic independence of certain formal power series (I), J. Sci. I. R. Iran, 2 (1991)50-55.
[10] —, Algebraic independence of certain formal power series (II), J. Sci. I. R. Iran, 3 (1992)148-
[11]—, Children products of formal power series, Math. Japonica, 38 (1993)319-324.
[12] —, E- algebraic functions over fields of positive characteristic- an analogue of differentially
algebraic functions, J. Algebra 207 (1998)355-366.
[13] —, Hadamard products of certain power series, Acta Arith. XCI (1999)95-105.
[14]—and C. F.Woodcock, Algebraic functions over a field of positive characteristic and Hadamard
products, J. London Math. Soc., 37 (1988) 395-403.
[15] K. Shikishima-Tsuji and M. Katsura, Hypertranscendental elements of a formal power series
ring of positive characteristic, Nagoya Math. J., 125 (1992)93-103.
[16] J.-Y. Yao, Some transcendental functions over function fields with positive characteristic, C.
R. Acad. Sci. Paris, Series I, 334 (2002)939-943.
[17] O. Zariski and P. Samuel, Commutative Algebra Vol. I, Van Nostrand, New York, 1958.