華林問題是去證明每一個自然數（足夠大）是一些有限範圍的正整數的ｄ次方的和；哥德巴赫問題是去證明每一個大過３的偶數是兩個質數的和，而每一個大過５的奇數是三個質數的和。所以，華林－哥德巴赫問題是探討把自然數展開成一些有限範圍的質數的ｄ次方（ｄ＞１）和的可能性。
本篇論文中，我們研究把有限體上的多項式展開成一些質多項式的ｄ次方（ｄ＞１）和的問題。這個問題和古典的華林－哥德巴赫問題相似，所以我們稱之為『多項式上的華林－哥德巴赫問題』。

section 1: Introduction
section 2: Exponential sums of polynomial rings
section 3: An estimation for sum involving the divisor function
section 4: Hua's lemma for Fq[T]
section 5: Vinogradov's mean value theorem for Fq[T]
section 6: Weyl's inequality for polynomial Waring problem
section 7: Hua's inequality for polynomial Waring problem
section 8: The minor and major arcs in polynomial
Waring-Goldbach problem
section 9: The polynomial Waring-Goldbach singular series
section 10:The polynomial Waring-Goldbach problem
section 11:A theorem in additive theory of irreducible
polynomials

1. L. Carlitz, The arithmetic of polynomials in a Galois field,
Am. J. Math, 54(1932), 39-50.
2. G. T. Diderrich and H. B. Mann, Representations by k-th
powers in GF(q), Journal of Number Theory, 4(1972), 269-273.
3. G. W. Effinger and D. R. Hayes, Additive Number Theory of
Polynomials Over a Finite Field, Oxford, Clarendon Press
(1991).
4. G. H. Hardy and E. W. Wright, An Introduction to the Theory
of Numbers, Oxford, Clarendon Press(1945).
5. D. R. Hayes, The expression of a polynomial as the sum of
three irreducibles, Acta Arith. 11(1966), 461-488.
6. C. N. Hsu, The Distribuction of Irreducible Polynomials in
Fq[t], Journal of Number Theory, 61(1996), 85-96.
7. C. N. Hsu, Diophantine Inequalities for Polynomial Rings,
to appear in Journal of Number Theory.
8. C. N. Hsu, On Hardy-Littlewood method for non-Archimedean
line, preprint.
9. L. K. Hua, On exponential sums, J. Chinese Math. Soc., 2
(1940), 301-312.
10.L. K. Hua, Additive Theory of Prime Numbers, American
Mathematical Society (1965).
11.R. Lidl and H. Niederreiter, Finite Field, Cambridge
University Press(1997).
12.M. B. Nathanson, Additive Number Theory (the classical
bases), GTM 164(1996).
13.M. B. Nathanson, Additive Number Theory (inverse problems
and the geometry of sumseys), GTM 165(1996).
14.S. Schwarz, On Waring's problem for finite fields, Quart.
Joun. Math, 19(1948), 123-128.
15.S. A. Stepanow, Arithmetic of Algebraic Curves, Translated
from Russian by Irene Aleksanova, Consultants Bureau, New
York(1994).
16.R. C. Vaughan , The Hardy-Littlewood method, Cambridge
University Press(1997).
17.W. A. Webb, Waring Problem in GF[q,x], Acta Arithmetica,
22(1973), 207-220.