Journal of Symbolic Computation最新文献

Given an ideal I in a polynomial ring $K[x_1,\dots ,x_n]$ over a field K, we present a complete algorithm to compute the binomial part of I, i.e., the subideal $\mathrm{Bin}\left(I\right)$ of I generated by all monomials and binomials in I. This is achieved step-by-step. First we collect and extend several algorithms for computing exponent lattices in different kinds of fields. Then we generalize them to compute exponent lattices of units in 0-dimensional K-algebras, where we have to generalize the computation of the separable part of an algebra to non-perfect fields in characteristic p. Next we examine the computation of unit lattices in finitely generated K-algebras, as well as their associated characters and lattice ideals. This allows us to calculate $\mathrm{Bin}\left(I\right)$ when I is saturated with respect to the indeterminates by reducing the task to the 0-dimensional case. Finally, we treat the computation of $\mathrm{Bin}\left(I\right)$ for general ideals by computing their cellular decomposition and dealing with finitely many special ideals called $(s,t)$-binomial parts. All algorithms have been implemented in SageMath.

Let η be a polarization with connected kernel on a superspecial abelian variety $E^g$. We give a sufficient criterion which allows the computation of the theta nullvalues of any quotient of $E^g$ by a maximal isotropic subgroup scheme of $\ker \left(\eta \right)$ effectively.

This criterion is satisfied in many situations studied by Li and Oort (1998). We used our method to implement an algorithm that computes supersingular curves of genus 3.

In this note, we precisely elaborate the connection between recognisable series (in the sense of Berstel and Reutenauer) and q-regular sequences (in the sense of Allouche and Shallit) via their linear representations. In particular, we show that the minimisation algorithm for recognisable series can also be used to minimise linear representations of q-regular sequences.

Twenty years after the discovery of the F5 algorithm, Gröbner bases with signatures are still challenging to understand and to adapt to different settings. This contrasts with Buchberger's algorithm, which we can bend in many directions keeping correctness and termination obvious. I propose an axiomatic approach to Gröbner bases with signatures with the purpose of uncoupling the theory and the algorithms, giving general results applicable in many different settings (e.g. Gröbner for submodules, F4-style reduction, noncommutative rings, non-Noetherian settings, etc.), and extending the reach of signature algorithms.

In this paper, we compute an algebraic decomposition of black-box rings in the generic ring model. More precisely, we explicitly decompose a black-box ring as a direct product of a nilpotent black-box ring and unital local black-box rings, by computing all its primitive idempotents. The algorithm presented in this paper uses quantum subroutines for the computation of the p-power parts of a black-box ring and then classical algorithms for the computation of the corresponding primitive idempotents. As a by-product, we get that the reduction of a black-box ring is also a black-box ring. The first application of this decomposition is an extension of the work of Maurer and Raub (2007) on representation problem in black-box finite fields to the case of reduced p-power black-box rings. Another important application is an ${\text{IND-CCA}}^1$ attack for any ring homomorphic encryption scheme in the generic ring model. Moreover, when the plaintext space is a finite reduced black-box ring, we present a plaintext-recovery attack based on representation problem in black-box prime fields. In particular, if the ciphertext space has smooth characteristic, the plaintext-recovery attack is effectively computable in the generic ring model.

Let G be a permutation group acting on a set Ω. Best known algorithms for computing the centralizer of G in the symmetric group on Ω are all based on the same general approach that involves solving the following two fundamental problems: given a G-orbit Δ of size n, compute the centralizer of the restriction of G to Δ in the symmetric group on Δ; and given two G-orbits Δ and ${\Delta }^{\prime }$ each of size n, find an equivalence between the action of G restricted to Δ and the action of G restricted to ${\Delta }^{\prime }$ when one exists. If G is given by a generating set X, previous solutions to each of these two problems take $O\left(\right|X\left|n^2\right)$ time.

In this paper, we first solve each fundamental problem in $O(\delta n+|X|n\log n)$ time, where δ is the depth of the breadth-first-search Schreier tree for X rooted at some fixed vertex. For the important class of small-base groups G, we improve the theoretical worst-case time bound of our solutions to $O(n{\log }^{c}n+|X|n\log n)$ for some constant c. Moreover, if $\lceil 20{\log }_{2}n\rceil$ uniformly distributed random elements of G are given in advance, our solutions have, with probability at least $1-1/n$, a running time of $O(n{\log }^{2}n+|X|n\log n)$. We then apply our solutions to obtain a more efficient algorithm for computing the centralizer of G in the symmetric group on Ω. In an experimental evaluation we demonstrate that it is substantially faster than previously known algorithms.

Satisfiability modulo theories (SMT) solvers check the satisfiability of quantifier-free first-order logic formulae over different theories. We consider the theory of non-linear real arithmetic where the formulae are logical combinations of polynomial constraints. Here a commonly used tool is the cylindrical algebraic decomposition (CAD) to decompose the real space into cells where the constraints are truth-invariant through the use of projection polynomials.

A CAD encodes more information than necessary for checking satisfiability. One approach to address this is to repackage the CAD theory into a search-based algorithm: one that guesses sample points to satisfy the formula, and generalizes guesses that conflict constraints to cylindrical cells around samples which are avoided in the continuing search. This can lead to a satisfying assignment more quickly, or conclude unsatisfiability with far fewer cells. A notable example of this approach is Jovanović and de Moura's NLSAT algorithm. Since these cells are being produced locally to a sample there is scope to use fewer projection polynomials than the traditional CAD projection. The original NLSAT algorithm reduced the set a little; while Brown's single cell construction reduced it much further still. However, it refines a cell polynomial-by-polynomial, meaning the shape and size of the cell produced depends on the order in which the polynomials are considered.

The present paper proposes a method to construct such cells levelwise, i.e. built level-by-level according to a variable ordering instead of polynomial-by-polynomial for all levels. We still use a reduced number of projection polynomials, but can now consider a variety of different reductions and use heuristics to select the projection polynomials in order to optimize the shape of the cell under construction. The new method can thus improve the performance of the NLSAT algorithm. We formulate all the necessary theory that underpins the algorithm as a proof system: while not a common presentation for work in this field, it is valuable in allowing an elegant decoupling of heuristic decisions from the main algorithm and its proof of correctness. We expect the symbolic computation community may find uses for it in other areas too. In particular, the proof system could be a step towards formal proofs for non-linear real arithmetic.

This work has been implemented in the SMT-RAT solver and the benefits of the levelwise construction are validated experimentally on the SMT-LIB benchmark library. We also compare several heuristics for the construction and observe that each heuristic has strengths offering potential for further exploitation of the new approach.

We propose a two-step Newton's method for refining an approximation of a singular zero whose deflation process terminates after one step, also known as a deflation-one singularity. Given an isolated singular zero of a square analytic system, our algorithm exploits an invertible linear operator obtained by combining the Jacobian and a projection of the Hessian in the direction of the kernel of the Jacobian. We prove the quadratic convergence of the two-step Newton method when it is applied to an approximation of a deflation-one singular zero. Also, the algorithm requires a smaller size of matrices than the existing methods, making it more efficient. We demonstrate examples and experiments to show the efficiency of the method.