Abstract
We prove the main conjecture in Vinogradov’s Mean Value Theorem for degrees higher than three. This will be a consequence of a sharp decoupling inequality for curves.

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@ARTICLE{BCT, mrkey = {2275834},
number = {2},
issn = {00015962},
author = {Bennett, Jonathan and Carbery, Anthony and Tao, Terence},
mrclass = {42B20 (44A15 45P05 46E35)},
doi = {10.1007/s1151100600064},
journal = {Acta Math.},
zblnumber = {1203.42019},
volume = {196},
mrnumber = {2275834},
fjournal = {Acta Mathematica},
mrreviewer = {Loukas Grafakos},
coden = {ACMAA8},
title = {On the multilinear restriction and {K}akeya conjectures},
year = {2006},
pages = {261302},
} 
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@ARTICLE{BCCT, mrkey = {2661170},
number = {4},
issn = {10732780},
author = {Bennett, Jonathan and Carbery, Anthony and Christ, Michael and Tao, Terence},
mrclass = {26D15},
doi = {10.4310/MRL.2010.v17.n4.a6},
journal = {Math. Res. Lett.},
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volume = {17},
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pages = {647666},
} 
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@misc{BBFL,
author = {Bennett, Jonathan and Bez, N. and Flock, T. and Lee, S.},
TITLE={Stability of {Br}ascamp{L}ieb constant and applications},
arxiv = {1508.07502},
} 
[Bo2] J. Bourgain, "Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces," Israel J. Math., vol. 193, iss. 1, pp. 441458, 2013.
@ARTICLE{Bo2, mrkey = {3038558},
number = {1},
issn = {00212172},
author = {Bourgain, Jean},
mrclass = {42B08 (42B05)},
doi = {10.1007/s1185601200771},
journal = {Israel J. Math.},
zblnumber = {1271.42039},
volume = {193},
mrnumber = {3038558},
fjournal = {Israel Journal of Mathematics},
mrreviewer = {F. M{ó}ricz},
coden = {ISJMAP},
title = {Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces},
year = {2013},
pages = {441458},
} 
[Bo] J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, 2014.
@misc{Bo,
author = {Bourgain, Jean},
TITLE = {Decoupling, exponential sums and the {R}iemann zeta function},
arxiv = {1408.5794},
YEAR={2014},
} 
[Bo6] J. Bourgain, Decoupling inequalities and some meanvalue theorems, 2014.
@misc{Bo6,
author = {Bourgain, Jean},
TITLE = {Decoupling inequalities and some meanvalue theorems},
NOTE={to appear in {\em J. d'Analyse Math.}},
arxiv = {1406.7862},
YEAR={2014},
} 
[BD3] J. Bourgain and C. Demeter, "The proof of the $l^2$ Decoupling Conjecture," Ann. of Math., vol. 182, iss. 1, pp. 351389, 2015.
@article {BD3, MRKEY = {3374964},
AUTHOR = {Bourgain, Jean and Demeter, Ciprian},
TITLE = {The proof of the {$l\sp 2$} {D}ecoupling {C}onjecture},
JOURNAL = {Ann. of Math.},
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VOLUME = {182},
YEAR = {2015},
NUMBER = {1},
PAGES = {351389},
ISSN = {0003486X},
MRCLASS = {42B37 (11E76 46E30 53C40)},
MRNUMBER = {3374964},
MRREVIEWER = {G. V. Rozenblum},
DOI = {10.4007/annals.2015.182.1.9},
zblnumber = {06456013},
} 
[BD6] J. Bourgain and C. Demeter, Mean value estimates for Weyl sums in two dimensions, 2015.
@misc{BD6,
author = {Bourgain, Jean and Demeter, Ciprian},
TITLE = {Mean value estimates for {W}eyl sums in two dimensions},
arxiv = {1509.05388},
year={2015},
} 
[BD4] J. Bourgain and C. Demeter, Decouplings for curves and hypersurfaces with nonzero Gaussian curvature, 2014.
@misc{BD4,
author = {Bourgain, Jean and Demeter, Ciprian},
TITLE = {Decouplings for curves and hypersurfaces with nonzero {G}aussian curvature},
NOTE={to appear in {\em J. d'Analyse Math.}},
arxiv = {1409.1634},
year={2014},
} 
[BD5] J. Bourgain and C. Demeter, "Decouplings for surfaces in $\mathbb{R}^4$," J. Funct. Anal., vol. 270, iss. 4, p. 1299–1318, 2016.
@article {BD5, MRKEY = {3447712},
AUTHOR = {Bourgain, Jean and Demeter, Ciprian},
TITLE = {Decouplings for surfaces in {$\mathbb{R}^4$}},
JOURNAL = {J. Funct. Anal.},
VOLUME = {270},
YEAR = {2016},
NUMBER = {4},
PAGES = {1299–1318},
ISSN = {00221236},
MRCLASS = {Preliminary Data},
MRNUMBER = {3447712},
DOI = {10.1016/j.jfa.2015.11.008},
zblnumber = {06535745},
} 
[BW] J. Bourgain and N. Watt, Decoupling for perturbed cones and mean square of $\zeta(\frac12+it)$, 2015.
@misc{BW,
author = {Bourgain, Jean and Watt, N.},
TITLE={Decoupling for perturbed cones and mean square of $\zeta(\frac12+it)$},
arxiv = {1505.04161},
year={2015},
} 
[BG] J. Bourgain and L. Guth, "Bounds on oscillatory integral operators based on multilinear estimates," Geom. Funct. Anal., vol. 21, iss. 6, pp. 12391295, 2011.
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AUTHOR = {Bourgain, Jean and Guth, Larry},
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FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {21},
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NUMBER = {6},
PAGES = {12391295},
ISSN = {1016443X},
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MRCLASS = {42B20},
MRNUMBER = {2860188},
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DOI = {10.1007/s0003901101409},
zblnumber = {1237.42010},
} 
[FoWo] K. Ford and T. D. Wooley, "On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing," Acta Math., vol. 213, iss. 2, pp. 199236, 2014.
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AUTHOR = {Ford, Kevin and Wooley, Trevor D.},
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FJOURNAL = {Acta Mathematica},
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NUMBER = {2},
PAGES = {199236},
ISSN = {00015962},
MRCLASS = {11P05},
MRNUMBER = {3286035},
MRREVIEWER = {Karin Halupczok},
DOI = {10.1007/s1151101401190},
zblnumber = {1307.11102},
} 
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FJOURNAL = {Proceedings of the Edinburgh Mathematical Society. Series II},
VOLUME = {52},
YEAR = {2009},
NUMBER = {3},
PAGES = {631651},
ISSN = {00130915},
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MRNUMBER = {2546636},
MRREVIEWER = {Fr{é}d{é}ric Bernicot},
DOI = {10.1017/S001309150700048X},
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} 
[Gu] L. Guth, "A short proof of the multilinear Kakeya inequality," Math. Proc. Cambridge Philos. Soc., vol. 158, iss. 1, pp. 147153, 2015.
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AUTHOR = {Guth, Larry},
TITLE = {A short proof of the multilinear {K}akeya inequality},
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FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
VOLUME = {158},
YEAR = {2015},
NUMBER = {1},
PAGES = {147153},
ISSN = {03050041},
MRCLASS = {42B08 (26D15)},
MRNUMBER = {3300318},
MRREVIEWER = {Timothy Michael Wertz},
DOI = {10.1017/S0305004114000589},
} 
[PrSe] M. Pramanik and A. Seeger, "$L^p$ regularity of averages over curves and bounds for associated maximal operators," Amer. J. Math., vol. 129, iss. 1, pp. 61103, 2007.
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AUTHOR = {Pramanik, Malabika and Seeger, Andreas},
TITLE = {{$L\sp p$} regularity of averages over curves and bounds for associated maximal operators},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {129},
YEAR = {2007},
NUMBER = {1},
PAGES = {61103},
ISSN = {00029327},
CODEN = {AJMAAN},
MRCLASS = {42B20},
MRNUMBER = {2288738},
MRREVIEWER = {Charles N. Moore},
DOI = {10.1353/ajm.2007.0003},
zblnumber = {1161.42009},
} 
[TWol] T. Wolff, "Local smoothing type estimates on $L^p$ for large $p$," Geom. Funct. Anal., vol. 10, iss. 5, pp. 12371288, 2000.
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AUTHOR = {Wolff, T.},
TITLE = {Local smoothing type estimates on {$L\sp p$} for large {$p$}},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {10},
YEAR = {2000},
NUMBER = {5},
PAGES = {12371288},
ISSN = {1016443X},
CODEN = {GFANFB},
MRCLASS = {42B25 (35L05 42B15)},
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MRREVIEWER = {Anthony Carbery},
DOI = {10.1007/PL00001652},
zblnumber = {0972.42005},
} 
[Wo] T. D. Wooley, "The cubic case of the main conjecture in Vinogradov’s mean value theorem," Adv. Math., vol. 294, pp. 532561, 2016.
@article{Wo,
author = {Wooley, Trevor D.},
TITLE = {The cubic case of the main conjecture in {V}inogradov's mean value theorem},
journal = {Adv. Math.},
fjournal = {Advances in Mathematics},
volume = {294},
pages = {532561},
year = {2016},
mrnumber = {3479572},
zblnumber = {06567880},
doi = {10.1016/j.aim.2016.02.033},
} 
[Wo1] T. D. Wooley, Approximating the main conjecture in Vinogradov’s mean value Theorem, 2014.
@misc{Wo1,
author = {Wooley, Trevor D.},
TITLE = {Approximating the main conjecture in {V}inogradov's mean value Theorem},
YEAR={2014},
arxiv = {1401.2932},
NOTE={52. pp.},
} 
@misc{Wo2,
author = {Wooley, Trevor D.},
TITLE = {Translation invariance, exponential sums, and {W}aring's problem},
SORTYEAR = {2014},
url ={http://www.icm2014.org/download/Proceedings_Volume_II.pdf},
note = {to appear in {\em Proc. Internat. Congress Math.},
August 1321 (Seoul, 2014), Volume II, Kyung Moon Sa Co. Ltd., Seoul, Korea pp. 505529},
} 
[Wo3] T. D. Wooley, "The asymptotic formula in Waring’s problem," Int. Math. Res. Not., vol. 2012, iss. 7, pp. 14851504, 2012.
@article {Wo3, MRKEY = {2913181},
AUTHOR = {Wooley, Trevor D.},
TITLE = {The asymptotic formula in {W}aring's problem},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices. IMRN},
YEAR = {2012},
NUMBER = {7},
PAGES = {14851504},
ISSN = {10737928},
MRCLASS = {11P05 (11P55)},
MRNUMBER = {2913181},
MRREVIEWER = {S. W. Graham},
doi = {10.1093/imrn/rnr074},
VOLUME = {2012},
zblnumber = {1267.11104},
} 
@misc{Wool1,
author = {Wooley, Trevor D.},
TITLE = {Discrete {F}ourier restriction via efficient congruencing},
SORTYEAR={2016},
note={{\em Internat. Math. Res. Notices},
48 pp.},
doi = {10.1093/imrn/rnw031},
}