TY - JOUR
AB - We establish the dimension and irreducibility of the moduli space of rational curves (of fixed degree) on arbitrary smooth hypersurfaces of sufficiently low degree. A spreading out argument reduces the problem to hypersurfaces defined over finite fields of large cardinality, which can then be tackled using a function field version of the Hardy-Littlewood circle method, in which particular care is taken to ensure uniformity in the size of the underlying finite field.
AU - Timothy Browning
AU - Vishe, Pankaj
ID - 265
IS - 7
JF - Geometric Methods in Algebra and Number Theory
TI - Rational curves on smooth hypersurfaces of low degree
VL - 11
ER -
TY - JOUR
AB - We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak approximation, and the Manin-Peyre conjecture for a smooth and geometrically integral variety X Pm, provided only that its dimension is large enough in terms of its degree.
AU - Timothy Browning
AU - Heath-Brown, Roger
ID - 266
IS - 2
JF - Journal of the European Mathematical Society
TI - Forms in many variables and differing degrees
VL - 19
ER -
TY - JOUR
AB - Building on recent work of Bhargava, Elkies and Schnidman and of Kriz and Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.
AU - Timothy Browning
ID - 267
IS - 3
JF - Mathematika
TI - Many cubic surfaces contain rational points
VL - 63
ER -
TY - JOUR
AB - We show that any subset of the squares of positive relative upper density contains nontrivial solutions to a translation-invariant linear equation in five or more variables, with explicit quantitative bounds. As a consequence, we establish the partition regularity of any diagonal quadric in five or more variables whose coefficients sum to zero. Unlike previous approaches, which are limited to equations in seven or more variables, we employ transference technology of Green to import bounds from the linear setting.
AU - Timothy Browning
AU - Prendiville, Sean M
ID - 268
IS - 7
JF - International Mathematics Research Notices
TI - A transference approach to a Roth-type theorem in the squares
VL - 2017
ER -
TY - JOUR
AU - Browning, Timothy D
AU - Loughran, Daniel
ID - 269
IS - 3-4
JF - Mathematische Zeitschrift
TI - Varieties with too many rational points
VL - 285
ER -
TY - JOUR
AB - Given a symmetric variety Y defined over Q and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski dense set of integral points on Y . We also establish an asymptotic counting formula for this set. In the special case that Y is a quadric hypersurface, we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics defined over Q.
AU - Timothy Browning
AU - Gorodnik, Alexander
ID - 270
IS - 6
JF - Proceedings of the London Mathematical Society
TI - Power-free values of polynomials on symmetric varieties
VL - 114
ER -
TY - JOUR
AB - We show that a non-singular integral form of degree d is soluble non-trivially over the integers if and only if it is soluble non-trivially over the reals and the p-adic numbers, provided that the form has at least (d-\sqrt{d}/2)2^d variables. This improves on a longstanding result of Birch.
AU - Timothy Browning
AU - Prendiville, Sean M
ID - 271
IS - 731
JF - Journal fur die Reine und Angewandte Mathematik
TI - Improvements in Birch's theorem on forms in many variables
VL - 2017
ER -
TY - JOUR
AB - Given a number field K/Q and a polynomial P ε Q [t], all of whose roots are Q, let X be the variety defined by the equation NK (x) = P (t). Combining additive combinatiorics with descent we show that the Brauer-Manin obstruction is the only obstruction to the Hesse principle and weak approximation on any smooth and projective model of X.
AU - Timothy Browning
AU - Matthiesen, Lilian
ID - 272
IS - 6
JF - Annales Scientifiques de l'Ecole Normale Superieure
TI - Norm forms for arbitrary number fields as products of linear polynomials
VL - 50
ER -
TY - CONF
AB - We consider the problem of estimating the partition function Z(β)=∑xexp(−β(H(x)) of a Gibbs distribution with a Hamilton H(⋅), or more precisely the logarithm of the ratio q=lnZ(0)/Z(β). It has been recently shown how to approximate q with high probability assuming the existence of an oracle that produces samples from the Gibbs distribution for a given parameter value in [0,β]. The current best known approach due to Huber [9] uses O(qlnn⋅[lnq+lnlnn+ε−2]) oracle calls on average where ε is the desired accuracy of approximation and H(⋅) is assumed to lie in {0}∪[1,n]. We improve the complexity to O(qlnn⋅ε−2) oracle calls. We also show that the same complexity can be achieved if exact oracles are replaced with approximate sampling oracles that are within O(ε2qlnn) variation distance from exact oracles. Finally, we prove a lower bound of Ω(q⋅ε−2) oracle calls under a natural model of computation.
AU - Kolmogorov, Vladimir
ID - 274
T2 - Proceedings of the 31st Conference On Learning Theory
TI - A faster approximation algorithm for the Gibbs partition function
VL - 75
ER -
TY - JOUR
AB - We present the fabrication and characterization of an aluminum transmon qubit on a silicon-on-insulator substrate. Key to the qubit fabrication is the use of an anhydrous hydrofluoric vapor process which selectively removes the lossy silicon oxide buried underneath the silicon device layer. For a 5.6 GHz qubit measured dispersively by a 7.1 GHz resonator, we find T1 = 3.5 μs and T∗2 = 2.2 μs. This process in principle permits the co-fabrication of silicon photonic and mechanical elements, providing a route towards chip-scale integration of electro-opto-mechanical transducers for quantum networking of superconducting microwave quantum circuits. The additional processing steps are compatible with established fabrication techniques for aluminum transmon qubits on silicon.
AU - Keller, Andrew J
AU - Dieterle, Paul
AU - Fang, Michael
AU - Berger, Brett
AU - Fink, Johannes M
AU - Painter, Oskar
ID - 796
IS - 4
JF - Applied Physics Letters
SN - 00036951
TI - Al transmon qubits on silicon on insulator for quantum device integration
VL - 111
ER -