Adjoint rings are finitely generated

This paper proves finite generation of the log canonical ring without Mori theory.Comment: completion of the project initiated in arXiv:0812.3046; v2,v3: presentation improved, includes parts of arXiv:0707.4414 in Sections 4 and

Factoring Safe Semiprimes with a Single Quantum Query

Shor's factoring algorithm (SFA), by its ability to efficiently factor large numbers, has the potential to undermine contemporary encryption. At its heart is a process called order finding, which quantum mechanics lets us perform efficiently. SFA thus consists of a \emph{quantum order finding algorithm} (QOFA), bookended by classical routines which, given the order, return the factors. But, with probability up to $1/2$, these classical routines fail, and QOFA must be rerun. We modify these routines using elementary results in number theory, improving the likelihood that they return the factors. The resulting quantum factoring algorithm is better than SFA at factoring safe semiprimes, an important class of numbers used in cryptography. With just one call to QOFA, our algorithm almost always factors safe semiprimes. As well as a speed-up, improving efficiency gives our algorithm other, practical advantages: unlike SFA, it does not need a randomly picked input, making it simpler to construct in the lab; and in the (unlikely) case of failure, the same circuit can be rerun, without modification. We consider generalizing this result to other cases, although we do not find a simple extension, and conclude that SFA is still the best algorithm for general numbers (non safe semiprimes, in other words). Even so, we present some simple number theoretic tricks for improving SFA in this case.Comment: v2 : Typo correction and rewriting for improved clarity v3 : Slight expansion, for improved clarit

Approximate computations with modular curves

This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations with modular curves and their Jacobians. These approximations are done in polynomial time in the dimension and the required number of significant digits. We explain the main ideas of how the approximations are done, illustrating them with examples, and we sketch some applications in number theory

On the minimal ranks of matrix pencils and the existence of a best approximate block-term tensor decomposition

Under the action of the general linear group with tensor structure, the ranks of matrices $A$ and $B$ forming an $m \times n$ pencil $A + \lambda B$ can change, but in a restricted manner. Specifically, with every pencil one can associate a pair of minimal ranks, which is unique up to a permutation. This notion can be defined for matrix pencils and, more generally, also for matrix polynomials of arbitrary degree. In this paper, we provide a formal definition of the minimal ranks, discuss its properties and the natural hierarchy it induces in a pencil space. Then, we show how the minimal ranks of a pencil can be determined from its Kronecker canonical form. For illustration, we classify the orbits according to their minimal ranks (under the action of the general linear group) in the case of real pencils with $m, n \le 4$. Subsequently, we show that real regular $2k \times 2k$ pencils having only complex-valued eigenvalues, which form an open positive-volume set, do not admit a best approximation (in the norm topology) on the set of real pencils whose minimal ranks are bounded by $2k-1$. Our results can be interpreted from a tensor viewpoint, where the minimal ranks of a degree-$(d-1)$ matrix polynomial characterize the minimal ranks of matrices constituting a block-term decomposition of an $m \times n \times d$ tensor into a sum of matrix-vector tensor products.Comment: This work was supported by the European Research Council under the European Programme FP7/2007-2013, Grant AdG-2013-320594 "DECODA.

A two variable Artin conjecture

Let a and b be non-zero rational numbers that are multiplicatively independent. We study the natural density of the set of primes p for which the subgroup of the multiplicative group of the finite field with p elements generated by (a\mod p) contains (b\mod p). It is shown that, under assumption of the generalized Riemann hypothesis (GRH), this density exists and equals a positive rational multiple of the universal constant S=\prod_{p prime}(1-p/(p^3-1)). An explicit value of the density is given under mild conditions on a and b. This extends and corrects earlier work of P.J. Stephens (1976). Our result, in combination with earlier work of the second author, allows us to deduce that any second order linear recurrence with reducible characteristic polynomial having integer elements, has a positive density of prime divisors (under GRH)

Scalability of Shor's algorithm with a limited set of rotation gates

Typical circuit implementations of Shor's algorithm involve controlled rotation gates of magnitude $\pi/2^{2L}$ where $L$ is the binary length of the integer N to be factored. Such gates cannot be implemented exactly using existing fault-tolerant techniques. Approximating a given controlled $\pi/2^{d}$ rotation gate to within $\delta=O(1/2^{d})$ currently requires both a number of qubits and number of fault-tolerant gates that grows polynomially with $d$. In this paper we show that this additional growth in space and time complexity would severely limit the applicability of Shor's algorithm to large integers. Consequently, we study in detail the effect of using only controlled rotation gates with $d$ less than or equal to some $d_{\rm max}$. It is found that integers up to length $L_{\rm max} = O(4^{d_{\rm max}})$ can be factored without significant performance penalty implying that the cumbersome techniques of fault-tolerant computation only need to be used to create controlled rotation gates of magnitude $\pi/64$ if integers thousands of bits long are desired factored. Explicit fault-tolerant constructions of such gates are also discussed.Comment: Substantially revised version, twice as long as original. Two tables converted into one 8-part figure, new section added on the construction of arbitrary single-qubit rotations using only the fault-tolerant gate set. Substantial additional discussion and explanatory figures added throughout. (8 pages, 6 figures

Enumerative Galois theory for cubics and quartics

We show that there are $O_\varepsilon(H^{1.5+\varepsilon})$ monic, cubic polynomials with integer coefficients bounded by $H$ in absolute value whose Galois group is $A_3$. We also show that the order of magnitude for $D_4$ quartics is $H^2 (\log H)^2$, and that the respective counts for $A_4$, $V_4$, $C_4$ are $O(H^{2.91})$, $O(H^2 \log H)$, $O(H^2 \log H)$. Our work establishes that irreducible non-$S_3$ cubic polynomials are less numerous than reducible ones, and similarly in the quartic setting: these are the first two solved cases of a 1936 conjecture made by van der Waerden