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  • Andreas O. Bender (a1)


A monic polynomial in ${\mathbf{F} }_{q} [t] $ of degree $n$ over a finite field ${\mathbf{F} }_{q} $ of odd characteristic can be written as the sum of two irreducible monic elements in ${\mathbf{F} }_{q} [t] $ of degrees $n$ and $n- 1$ if $q$ is larger than a bound depending only on $n$ . The main tool is a sufficient condition for simultaneous primality of two polynomials in one variable $x$ with coefficients in ${\mathbf{F} }_{q} [t] $ .



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1.Bender, A. O., Decompositions into sums of two irreducibles in ${\mathbf{F} }_{q} [t] $. C. R. Math. Acad. Sci. Paris 346 (17–18) (2008), 931934; also available from
2.Bender, A. O. and Wittenberg, O., A potential analogue of Schinzel’s hypothesis for polynomials with coefficients in ${\mathbf{F} }_{q} [t] $. Int. Math. Res. Not. 36 (2005), 22372248; also available from
3.Car, M., Le problème de Goldbach pour l’anneau des polynômes sur un corps fini. C. R. Acad. Sci. Paris, Sér. A-B 273 (1971), A201A204.
4.Car, M., Le théorème de Chen pour ${\mathbf{F} }_{q} [X] $. Dissertationes Math. (Rozprawy Mat.) 223 (1984), Polska Akademia Nauk. Instytut Matematyczny.
5.Car, M., The generalized polynomial Goldbach problem. J. Number Theory 57 (1) (1996), 2249.
6.Chen, J.-R., On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Sci. Sinica 16 (2) (1973), 157176; also available in Y. Wang (ed.), The Goldbach Conjecture, 2nd edn., World Scientific Publishing (Singapore, 2002).
7.Cherly, J., A lower bound theorem in ${\mathbf{F} }_{q} [x] $. J. Reine Angew. Math. 303/304 (1978), 253264.
8.Davenport, H., Multiplicative Number Theory, 3rd edn. (Graduate Texts in Mathematics 74), Springer (New York, NY, 2000), revised by Hugh L. Montgomery.
9.Effinger, G. W. and Hayes, D. R., A complete solution to the polynomial 3-primes problem. Bull. Amer. Math. Soc. (N.S.) 24 (2) (1991), 363369.
10.Effinger, G. W. and Hayes, D. R., Additive Number Theory of Polynomials over a Finite Field, Oxford University Press (New York, NY, 1991).
11.Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry (Graduate Texts in Mathematics 150), Springer (New York, NY, 1995).
12.Euler, L., Letter to Christian Goldbach dated 30th June 1742. In Leonhard Euler und Christian Goldbach: Briefwechsel 1729–1764 (eds Juskevic, A. P. and Winter, E.),Akademie (Berlin, 1965), also available as Lettre XLIV from
13.Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S. L., Nitsure, N. and Vistoli, A., Fundamental Algebraic Geometry, Grothendieck’s fga explained (Mathematical Surveys and Monographs 123), American Mathematical Society (Providence, RI, 2005).
14.Gelfand, I. M., Kapranov, M. and Zelevinsky, A., Discriminants, Resultants and Multidimensional Determinants, Birkhäuser (Boston, MA, 1994).
15.Geyer, W.-D. and Jarden, M., Bounded realization of $l$-groups over global fields. Nagoya Math. J. 150 (1998), 1362.
16.Goldbach, C., Letter to Leonhard Euler dated 7th June 1742. In Leonhard Euler und Christian Goldbach: Briefwechsel 1729–1764 (eds Juskevic, A. P. and Winter, E.),Akademie (Berlin, 1965),
17.Grothendieck, A., Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): IV. Étude locale des schémas et des morphismes de schémas, II. Publ. Math. Inst. Hautes Études Sci. 24 (1965), 5231.
18.Hayes, D. R., A polynomial analog of the Goldbach conjecture. Bull. Amer. Math. Soc. 69 (1963), 115116; Correction ibid. 493.
19.Homma, M., Funny plane curves in characteristic $p\gt 0$. Comm. Algebra 15 (7) (1987), 14691501.
20.Hurwitz, A., Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten. Math. Ann. 39 (1891), 160; and Math. Werke, Band 1/XXI, Birkhäuser (Basel, 1932).
21.Jouanolou, J.-P., Le formalisme du résultant. Adv. Math. 90 (2) (1991), 117263.
22.Lorenzini, D., An Invitation to Arithmetic Geometry, American Mathematical Society (Providence, RI, 1996).
23.Nathanson, M. B., Additive Number Theory: the Classical Bases (Graduate Texts in Mathematics 164), Springer (New York, NY, 1996).
24.Pollack, P., Prime polynomials over finite fields, PhD Thesis, Dartmouth College, 2008.
25.Pollack, P., The exceptional set in the polynomial Goldbach problem. Int. J. Number Theory 7 (3) (2011), 579591.
26.Schmidt, W. M., Equations Over Finite Fields: An Elementary Approach (Lecture Notes in Mathematics 536), Springer (Heidelberg, 1976).
27.Serre, J.-P., Topics in Galois Theory, Jones and Bartlett Publishers (Boston, MA, 1992).
28.Vinogradov, I. M., Representation of an odd number as a sum of three primes. Dokl. Akad. Nauk SSSR 15 (6–7) (1937), 291294.
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  • Andreas O. Bender (a1)


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