J. Org, O. Br¨udern, and . Robert, If there are 2S 2,e , then there is also a least one S 2,o , and as in case (?), this odd variable can be used to ensure one S 1,e . But then we complete the argument via

, Now ? 1 ? 6. The more frequent parity of the variables at niveau 1 occurs at least three times, and two of them contract to a second S 2,e . This leaves four variables at niveau 1, and by using one of the S 2,o if necessary, we can ensure that we have an S 1,e available, This leaves the case ? 2 = 3 We can now complete the argument via

, This is similar to case (v), but there are certain details that require attention We begin with (16.1) and (16.2), providing 2P 1,e , P 0,o , P 0,e or 3P 1,e . If ? 1 ? 8, then Lemma 14.3 again yields a P 4,e . If ? 1 = 7 and there is a variable S j,o with j ? 2, then use (13.12), so that we have 3P 1,e available, Otherwise, all variables at niveaux 2 and 3 are even

, implies ? 2 ? 3, providing 3S 2,e , and 2P 1,e ? P 2

, We now follow the argument given in case (v)

, Here, as above (17.4) 2P 1,e , 3S 2,e ? P 2,e , 3S 2,e ? P 4,e . completes the argument. (?) Suppose that there are 3S 2,o . These contract to S 3,e , S 2,o , and the remaining S 2,o can be used in (13.12) to ensure that we have 3P 1,e . If there is an S 1,e , then (17.2) yields a P 4,e . In the alternative case, we have at least 3S 1,o , providing an S 2

, If the system is not covered by (?) or (?), we see from ? 2 ? 4 that we must have ? 2 = 4 with 2S 2,o , 2S 2,e . But now ? 1 = 6, and as in case (v), one then may construct an S 2,e from the variables

, Here (16.1) yields 2P 1,e . If ? 1 ? 8 then Lemma 14

. Hence,

, If there are 3S 2,e , we use (17.4)

, If there are 3S 2,o , transform theses to S 2,o , S 3,e . Should there be

, Now (17.5) completes the argument. (?) If the system is not covered by (?) or (?), then ? 2 = 4, with 2S 2,e , 2S 2,o . We use the 2S 2,o to ensure 2S 1,e at niveau 1, and then

, The proof if Lemma 17.2 is now complete

G. I. Arkhipov and A. A. Karatsuba, Local representation of zero by a form. (Russian) Izv. Akad, Nauk SSSR Ser. Mat, vol.45, issue.5, pp.948-961, 1981.

J. Ax and S. Kochen, Diophantine Problems Over Local Fields I, American Journal of Mathematics, vol.87, issue.3, pp.605-630, 1965.
DOI : 10.2307/2373065

D. Brownawell, On p-adic zeros of forms, Journal of Number Theory, vol.18, issue.3, pp.342-349, 1984.
DOI : 10.1016/0022-314X(84)90066-0

URL : https://doi.org/10.1016/0022-314x(84)90066-0

J. Brüdern and H. Godinho, On Artin's Conjecture, I: Systems of Diagonal Forms, Bulletin of the London Mathematical Society, vol.31, issue.3, pp.305-313, 1999.
DOI : 10.1112/S0024609398005578

J. Brüdern and H. Godinho, On Artin's Conjecture, II: Pairs of Additive Forms, Proc. London Math. Soc. (3), pp.513-538, 2002.
DOI : 10.1112/S0024611502013588

S. Chowla, H. B. Mann, and E. G. Straus, Some applications of the Cauchy-Davenport theorem . Norske Vid, Selsk. Forh. Trondheim, vol.32, pp.74-80, 1959.

H. Davenport, Cubic Forms in Thirty-Two Variables, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.251, issue.993, pp.193-232, 1959.
DOI : 10.1098/rsta.1959.0002

H. Davenport and D. J. Lewis, Homogeneous Additive Equations, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.274, issue.1359, pp.443-460, 1963.
DOI : 10.1098/rspa.1963.0143

H. Davenport and D. J. Lewis, Cubic Equations of Additive Type, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.261, issue.1117, pp.97-136, 1966.
DOI : 10.1098/rsta.1966.0060

H. Davenport and D. J. Lewis, Simultaneous Equations of Additive Type, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.264, issue.1155, pp.557-595, 1969.
DOI : 10.1098/rsta.1969.0035

H. Davenport and D. J. Lewis, Two additive equations, Number Theory (Proc. Sympos, pp.74-98, 1967.
DOI : 10.1090/pspum/012/0253985

V. B. , Dem'yanov, On cubic forms in discretely normed fields. (Russian) Doklady Akad, Nauk SSSR (N.S.), vol.74, pp.889-891, 1950.

M. Dodson, Homogeneous Additive Congruences, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.261, issue.1119, pp.163-210, 1967.
DOI : 10.1098/rsta.1967.0002

J. H. , Dumke, p-adic zeros of quintic forms

W. J. Ellison, A 'Waring Problem' for homogeneous forms, Proc. Cambridge Philos, pp.663-672, 1969.

M. P. Knapp, Pairs of additive forms of odd degrees. Michigan Math, J, vol.61, issue.3, pp.493-505, 2012.

C. Kränzlein, Paare additiver Formen vom Grad 2 n . Dissertation, 2009.

D. J. Lewis, Cubic Homogeneous Polynomials Over ???-Adic Number Fields, The Annals of Mathematics, vol.56, issue.3, pp.56-473, 1952.
DOI : 10.2307/1969655

D. J. Lewis and H. L. Montgomery, On zeros of p-adic forms. Michigan Math, J, vol.30, issue.1, pp.83-87, 1983.

L. Low, J. Pitman, and A. Wolff, Simultaneous diagonal congruences, Journal of Number Theory, vol.29, issue.1, pp.31-59, 1988.
DOI : 10.1016/0022-314X(88)90092-3

A. Meyer, Mathematische Mittheilungen, Vierteljahrschrift der Naturforschenden Gesellschaft in Zürich, vol.29, pp.209-222, 1884.

M. B. Nathanson, Additive number theory Inverse problems and the geometry of sumsets, Graduate Texts in Mathematics, vol.165, 1996.

J. E. Olson, A combinatorial problem on finite Abelian groups, I, Journal of Number Theory, vol.1, issue.1, pp.8-10, 1969.
DOI : 10.1016/0022-314X(69)90021-3

B. Reznick, On the length of binary forms Quadratic and higher degree forms, Dev. Math, vol.31, pp.207-232

G. Terjanian, Un contre-exemplè a une conjecture d'Artin, French) C. R. Acad. Sci. Paris Sér. A-B 262, p.612, 1966.

G. Terjanian, Formes p-adiques anisotropes. (French) J. Reine Angew, Math, pp.313-217, 1980.

T. D. Wooley, On simultaneous additive equations. I, Proc. London Math. Soc. (3), pp.1-34, 1991.
DOI : 10.1112/plms/s3-63.1.1

T. D. Wooley, Abstract, Forum Mathematicum, vol.27, issue.4, pp.2259-2265, 2015.
DOI : 10.1515/forum-2013-0109

T. D. Wooley, Diophantine problems in many variables: the role of additive number theory Topics in number theory, Math. Appl, vol.467, pp.49-83, 1997.

, Jörg Brüdern Universität Göttingen Mathematisches Institut Bunsenstrasse, pp.3-5

G. Germany-bruedern@uni-math,