Computer Science

Given a nonsingular n×n matrix of univariate polynomials over a field K , we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use O ˜ ( n ω ⌈s⌉) operations in K , where s is bounded from above by both the average of the degrees of the rows and that of the columns of the matrix and ω is the exponent of matrix multiplication. The soft- O notation indicates that logarithmic factors in the big- O are omitted while the ceiling function indicates that the cost is O ˜ ( n ω ) when s=o(1) . Our algorithms are based on a fast and deterministic triangularization method for computing the diagonal entries of the Hermite form of a nonsingular matrix.

In this paper, we propose new deterministic and Monte Carlo interpolation algorithms for sparse multivariate polynomials represented by straight-line programs. Let f be an n -variate polynomial given by a straight-line program, which has a degree bound D and a term bound T . Our deterministic algorithm is quadratic in n,T and cubic in logD in the Soft-Oh sense, which has better complexities than existing deterministic interpolation algorithms in most cases. Our Monte Carlo interpolation algorithms have better complexities than existing Monte Carlo interpolation algorithms and are the first algorithms whose complexities are linear in nT in the Soft-Oh sense. Since nT is a factor of the size of f , our Monte Carlo algorithms are optimal in n and T in the Soft-Oh sense.

We set new speed records for multiplying long polynomials over finite fields of characteristic two. Our multiplication algorithm is based on an additive FFT (Fast Fourier Transform) by Lin, Chung, and Huang in 2014 comparing to previously best results based on multiplicative FFTs. Both methods have similar complexity for arithmetic operations on underlying finite field; however, our implementation shows that the additive FFT has less overhead. For further optimization, we employ a tower field construction because the multipliers in the additive FFT naturally fall into small subfields, which leads to speed-ups using table-lookup instructions in modern CPUs. Benchmarks show that our method saves about 40% computing time when multiplying polynomials of 2 28 and 2 29 bits comparing to previous multiplicative FFT implementations.

The Butler-Portugal algorithm for obtaining the canonical form of a tensor expression with respect to slot symmetries and dummy-index renaming suffers, in certain cases with a high degree of symmetry, from O(n!) explosion in both computation time and memory. We present a modified algorithm which alleviates this problem in the most common cases---tensor expressions with subsets of indices which are totally symmetric or totally antisymmetric---in polynomial time. We also present an implementation of the label-renaming mechanism which improves upon that of the original Butler-Portugal algorithm, thus providing a significant speed increase for the average case as well as the highly-symmetric special case. The worst-case behavior remains O(n!) , although it occurs in more limited situations unlikely to appear in actual computations. We comment on possible strategies to take if the nature of a computation should make these situations more likely.

We present an algorithm that computes the product of two n-bit integers in O(n log n (4\sqrt 2)^{log^* n}) bit operations. Previously, the best known bound was O(n log n 6^{log^* n}). We also prove that for a fixed prime p, polynomials in F_p[X] of degree n may be multiplied in O(n log n 4^{log^* n}) bit operations; the previous best bound was O(n log n 8^{log^* n}).

Assuming a conjectural upper bound for the least prime in an arithmetic progression, we show that n-bit integers may be multiplied in O(n log n 4^(log^* n)) bit operations.

We prove that n -bit integers may be multiplied in O(nlogn 4 log ∗ n ) bit operations. This complexity bound had been achieved previously by several authors, assuming various unproved number-theoretic hypotheses. Our proof is unconditional, and depends in an essential way on Minkowski's theorem concerning lattice vectors in symmetric convex sets.

We present new algorithms for computing the low n bits or the high n bits of the product of two n-bit integers. We show that these problems may be solved in asymptotically 75% of the time required to compute the full 2n-bit product, assuming that the underlying integer multiplication algorithm relies on computing cyclic convolutions of real sequences.

Given a multi-variant polynomial inequality with a parameter, how to find the best possible value of this parameter that satisfies the inequality? For instance, find the greatest number k that satisfies a 3 + b 3 + c 3 +k( a 2 b+ b 2 c+ c 2 a)−(k+1)(a b 2 +b c 2 +c a 2 )≥0 for all nonnegative real numbers a,b,c . Analogues problems often appeared in studies of inequalities and were dealt with by various methods. In this paper, a general algorithm is proposed for finding the required best possible constant. The algorithm can be easily implemented by computer algebra tools such as Maple.

Motivated by problems arising with the symbolic analysis of steady state ideals in Chemical Reaction Network Theory, we consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a coset of a multiplicative group. That property corresponds to Shifted Toricity, a recent generalization of toricity of the corresponding polynomial ideal. The key idea is to take a geometric view on varieties rather than an algebraic view on ideals. Recently, corresponding coset tests have been proposed for complex and for real varieties. The former combine numerous techniques from commutative algorithmic algebra with Gröbner bases as the central algorithmic tool. The latter are based on interpreted first-order logic in real closed fields with real quantifier elimination techniques on the algorithmic side. Here we take a new logic approach to both theories, complex and real, and beyond. Besides alternative algorithms, our approach provides a unified view on theories of fields and helps to understand the relevance and interconnection of the rich existing literature in the area, which has been focusing on complex numbers, while from a scientific point of view the (positive) real numbers are clearly the relevant domain in chemical reaction network theory. We apply prototypical implementations of our new approach to a set of 129 models from the BioModels repository.