Journal of Symbolic Computation最新文献

A multistationarity region is the part of a reaction network's parameter space that gives rise to multiple steady states. Mathematically, this region consists of the positive parameters for which a parametrized family of polynomial equations admits two or more positive roots. Much recent work has focused on analyzing multistationarity regions of biologically significant reaction networks and determining whether such regions are connected; indeed, a better understanding of the topology and geometry of such regions may help elucidate how robust multistationarity is to perturbations. Here we focus on the multistationarity regions of small networks, those with few species and few reactions. For two families of such networks – those with one species and up to three reactions, and those with two species and up to two reactions – we prove that the resulting multistationarity regions are connected. We also give an example of a network with one species and six reactions for which the multistationarity region is disconnected. Our proofs rely on the formula for the discriminant of a trinomial, a classification of small multistationary networks, and a recent result of Feliu and Telek that partially generalizes Descartes' rule of signs.

In 1936, Krull asked if the integral closure of a primary ideal is still primary. Fifty years later, Huneke partially answered this question by giving a primary polynomial ideal whose integral closure is not primary in a regular local ring of characteristic $p=2$. We provide counterexamples to Krull's question regarding polynomial rings over any fields. We also find that the Jacobian ideal J of the polynomial $f=x^6+y^6+x^{4}zt+z^3$ given by Briançon and Speder (1975) is a counterexample to Krull's question.

We present a computational approach to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table. We also present an example of a Lie algebra for which the quotient algebra of the almost-inner derivations modulo the inner derivations is non-abelian. This answers a question of Kunyavskii and Ostapenko.

Recovery of multivariate exponential polynomials, i.e., the multivariate version of Prony's problem, can be stabilized by using more than the minimally needed multiinteger samples of the function. We present an algorithm that takes into account this extra information and prove a backward error estimate for the algebraic recovery method SMILE. In addition, we give a method to approximate data by an exponential polynomial sequence of a given structure as a step in the direction of multivariate model reduction.

We explore algorithmic aspects of a simply transitive commutative group action coming from the class field theory of imaginary hyperelliptic function fields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over $F_q$ acts on a subset of isomorphism classes of Drinfeld modules. We describe an algorithm to compute the group action efficiently. This is a function field analog of the Couveignes-Rostovtsev-Stolbunov group action. We report on an explicit computation done with our proof-of-concept C++/NTL implementation; it took a fraction of a second on a standard computer. We prove that the problem of inverting the group action reduces to the problem of finding isogenies of fixed τ-degree between Drinfeld $F_q\left[X\right]$-modules, which is solvable in polynomial time thanks to an algorithm by Wesolowski. We give asymptotic complexity bounds for all algorithms presented in this paper.

Consider the symmetric group $S_n$ acting as a reflection group on the polynomial ring $k[x_1,\dots ,x_n]$ where k is a field, such that Char(k) does not divide n!. We use Higher Specht polynomials to construct matrix factorizations of the discriminant of this group action: these matrix factorizations are indexed by partitions of n and respect the decomposition of the coinvariant algebra into isotypical components. The maximal Cohen–Macaulay modules associated to these matrix factorizations give rise to a noncommutative resolution of the discriminant and they correspond to the nontrivial irreducible representations of $S_n$. All our constructions are implemented in Macaulay2 and we provide several examples. We also discuss extensions of these results to Young subgroups of $S_n$ and indicate how to generalize them to the reflection groups $G(m,1,n)$.

We propose a new algorithm to compute the topology of a real algebraic space curve. The novelties of this algorithm are a new technique to achieve the lifting step which recovers points of the space curve in each plane fiber from several projections and a weaker notion of generic position. As distinct to previous work, our sweep generic position does not require that x-critical points have different x-coordinates. The complexity of achieving this sweep generic position property is thus no longer a bottleneck in term of complexity. The bit complexity of our algorithm is $\tilde{O}(d^{18}+d^{17}\tau )$ where d and τ bound the degree and the bitsize of the integer coefficients, respectively, of the defining polynomials of the curve and polylogarithmic factors are ignored. To the best of our knowledge, this improves upon the best currently known results at least by a factor of $d^2$.

Inspired by Karr's algorithm, we consider the summations involving a sequence satisfying a recurrence of order two. The structure of such summations provides an algebraic framework for solving the difference equations of form $a\sigma \left(g\right)+bg=f$ in the bivariate difference field $\left(F\right(\alpha ,\beta ),\sigma )$, where $a,b,f\in F(\alpha ,\beta )\setminus \left\{0\right\}$ are known binary functions of α, β, and α, β are two algebraically independent transcendental elements, σ is a transformation that satisfies $\sigma \left(\alpha \right)=\beta$, $\sigma \left(\beta \right)=u\alpha +v\beta$, where $u,v\ne 0\in F$. Based on it, we then describe algorithms for finding the universal denominator for those equations in the bivariate difference field under certain assumptions. This reduces the general problem of finding the rational solutions of such equations to the problem of finding the polynomial solutions of such equations.

Tensors are ubiquitous in mathematics and the sciences, as they allow to store information in a concise way. Decompositions of tensors may give insights into their structure and complexity. In this work, we develop a new framework for decompositions of tensors, taking into account invariance, positivity and a geometric arrangement of their local spaces. We define an invariant decomposition with indices arranged on a simplicial complex which is explicitly invariant under a group action. We give a constructive proof that this decomposition exists for all invariant tensors, after possibly enriching the simplicial complex. We further define several decompositions certifying positivity, and prove similar existence results, as well as inequalities between the corresponding ranks. Our results generalize results from the theory of tensor networks, used in the study of quantum many-body systems, for example.