| 1. |
- Brandes, Julia, 1986
(author)
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Linear spaces on hypersurfaces over number fields
- 2017
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In: Michigan Mathematical Journal. - : Michigan Mathematical Journal. - 1945-2365 .- 0026-2285. ; 66:4, s. 769-784
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Journal article (peer-reviewed)abstract
- We establish an analytic Hasse principle for linear spaces of affine dimension m on a complete intersection over an algebraic field extension K of Q. The number of variables required to do this is no larger than what is known for the analogous problem over ?. As an application, we show that any smooth hypersurface over K whose dimension is large enough in terms of the degree is K-unirational, provided that either the degree is odd or K is totally imaginary.
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| 2. |
- Brandes, Julia, 1986, et al.
(author)
-
Simultaneous Additive Equations: Repeated and Differing Degrees
- 2017
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In: Canadian Journal of Mathematics. - 1496-4279 .- 0008-414X. ; 69:2, s. 258-283
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Journal article (peer-reviewed)abstract
- We obtain bounds for the number of variables required to establish Hasse principles, both for the existence of solutions and for asymptotic formula, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning-Heath-Brown, which give weak results when specialized to the diagonal situation, this is the first result on such "hybrid" systems. We also obtain specialized results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and r quadratic equations.
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| 3. |
- Brandes, Julia
(author)
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The Hasse Principle for Systems of Quadratic and Cubic Diagonal Equations
- 2017
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In: Quarterly Journal of Mathematics. - : Oxford University Press (OUP). - 0033-5606 .- 1464-3847. ; 68:3, s. 831-850
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Journal article (peer-reviewed)abstract
- Employing Brudern's and Wooley's new complification method, we establish an asymptotic Hasse principle for the number of solutions to a system of r(3) cubic and r(2) quadratic diagonal forms, where r3 > 2r2 > 0, in s > 6r3 + 143r2+ 1variables.
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| 4. |
- Brandes, Julia, 1986, et al.
(author)
-
Vinogradov systems with a slice off
- 2017
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In: Mathematika. - 0025-5793 .- 2041-7942. ; 63:3, s. 797-817
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Journal article (peer-reviewed)abstract
- Let I-s,I-k,I-r(X) denote the number of integral solutions of the modified Vinogradov system of equations with 1 <= x(i),y(i) <= X (1 <= i <= s). By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for I-s,I-k,I-r(X) for 1 <= r <= k - 1. In particular, when s,k is an element of N satisfy k >= 3 and 1 <= s <= (k(2) - 1)/2, we establish the essentially diagonal behaviour I-s,I-k,I-l(X) << Xs+epsilon.
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