We consider a system of @math cubic forms in @math variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided @math , we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular we can handle systems of forms in @math variables, previous work having required that @math . One conjectures that @math should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.
Br "udern and Wooley @cite_0 @cite_7 @cite_13 treat diagonal systems in @math variables. This is the best value of @math possible with the classical circle method. In particular they prove the Hasse principle for @math whenever the @math are diagonal, @math is smooth and @math . They also prove an asymptotic formula of the type whenever @math holds, or when @math and @math holds @cite_11 @cite_8 @cite_15 . In the case @math they prove a Hasse principle for certain pairs of diagonal cubics in as few as 11 variables @cite_9 .
@cite_7 @cite_8 @cite_9 @cite_0 @cite_15 @cite_13 @cite_11
1570283293 2048050639 2035417470 2133160967 2606394847 2949393806 1971773339
We establish the Hasse principle for systems of r simultaneous diagonal cubic equations whenever the number of variables exceeds 6r and the associated coefficient matrix contains no singular (r r ) submatrix, thereby achieving the theoretical limit of the circle method for such systems. This paper concerns systems of r homogeneous diagonal equations of degree k in s variables, with integer coefficients. Subject to a suitable non-singularity condition, it is shown that the expected asymptotic formula holds for the number of such systems inside a box [− P , P ] s , provided only that s > (3 r +1)2 k −2 . By way of comparison, classical methods based on the use of Hua's lemma would establish a similar conclusion, provided instead that s > r 2 k . Consider a system of diagonal equations satisfying the property that the (fixed) integral coefficient matrix @math contains no singular @math submatrix. A recent paper of the authors [ 3 ] establishes that whenever @math and @math , then the expected asymptotic formula holds for the number @math of integral solutions @math of ( @math ) with @math @math . By means of the Hardy-Littlewood method, we apply a new mean value theorem for exponential sums to confirm the truth, over the rational numbers, of the Hasse principle for pairs of diagonal cubic forms in thirteen or more variables. We develop several efficient algorithms for the classical problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input @math matrix @math , this problem asks to find diagonal (scaling) matrices @math and @math (if they exist), so that @math @math -approximates a doubly stochastic, or more generally a matrix with prescribed row and column sums. We address the general scaling problem as well as some important special cases. In particular, if @math has @math nonzero entries, and if there exist @math and @math with polynomially large entries such that @math is doubly stochastic, then we can solve the problem in total complexity @math . This greatly improves on the best known previous results, which were either @math or @math . Our algorithms are based on tailor-made first and second order techniques, combined with other recent advances in continuous optimization, which may be of independent interest for solving similar problems. We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most worse than the relaxation's value (normalized to lie in [0,1]), as long as r > k ( ) ( ) ;, where k is the alphabet size of I, @math , and @math denotes the number of eigenvalues larger than @math in the normalized adjacency matrix of the constraint graph of @math . In the case that @math is a instance, the threshold @math is only a polynomial in @math , and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture. Our algorithm actually requires less than the @math constraints specified by the @math level of the Lasserre hierarchy, and in some cases @math rounds of our program can be evaluated in time @math . Consider a network of @math n processes, where each process inputs a @math d-dimensional vector of reals. All processes can communicate directly with others via reliable FIFO channels. We discuss two problems. The multidimensional Byzantine consensus problem, for synchronous systems, requires processes to decide on a single @math d-dimensional vector @math v?Rd, inside the convex hull of @math d-dimensional vectors that were input by the non-faulty processes. Also, the multidimensional Byzantine approximate agreement (MBAA) problem, for asynchronous systems, requires processes to decide on multiple @math d-dimensional vectors in @math Rd, all within a fixed Euclidean distance @math ∈ of each other, and inside the convex hull of @math d-dimensional vectors that were input by the non-faulty processes. We obtain the following results for the problems above, while tolerating up to @math f Byzantine failures in systems with complete communication graphs: (1) In synchronous systems, @math n>max 3f,(d+1)f is necessary and sufficient to solve the multidimensional consensus problem. (2) In asynchronous systems, @math n>(d+2)f is necessary and sufficient to solve the multidimensional approximate agreement problem. Our sufficiency proofs are constructive, giving explicit protocols for the problems. In particular, for the MBAA problem, we give two protocols with strictly different properties and applications.