Using Kapur-Sun-Wang algorithm for the Gröbner Cover

Kapur-Sun-Wang have recently developed a very efficient algorithm for computing Comprehensive Gröbner Systems that has moreover the required essential properties for being used as first step of the Gröbner Cover algorithm. We have implemented and adapted it inside the Singular grobcov library for computing the Gröbner Cover and there are evidences that it makes the canonical algorithm much more effective. In this note we discuss the performance of GC with KSW on a collection of examples.Peer ReviewedPostprint (published version

Thomas Decomposition of Algebraic and Differential Systems

In this paper we consider disjoint decomposition of algebraic and non-linear partial differential systems of equations and inequations into so-called simple subsystems. We exploit Thomas decomposition ideas and develop them into a new algorithm. For algebraic systems simplicity means triangularity, squarefreeness and non-vanishing initials. For differential systems the algorithm provides not only algebraic simplicity but also involutivity. The algorithm has been implemented in Maple

Solving Parametric Polynomial Systems

We present a new algorithm for solving basic parametric constructible or semi-algebraic systems like \mathcal{C} = \{ x \in \Cp_1 ( x ) = 0, \ldots, p_s ( x ) = 0, f_1 ( x ) \neq 0, \ldots, f_l ( x ) \neq 0 \} or \mathcal{S} = \{ x \in \Cp_1 ( x ) = 0, \ldots, p_s ( x ) = 0, f_1 ( x ) > 0, \ldots, f_l ( x ) > 0 \}, where p_i, f_i \in \Q [ U, X ], $U = [ U_1, \ldots, U_d ]$ is the set of parameters and $X = [ X_{d + 1}, \ldots, X_n ]$ the set of unknowns. If $\Pi_U$ denotes the canonical projection on the parameter's space, solving $\mathcal{C}$ or $\mathcal{S}$ remains to compute sub-manifolds \mathcal{U} \subset \C(resp. \mathcal{U} \subset \R^d$) such that$( \Pi_U^{- 1} ( \mathcal{U} ) \cap \mathcal{C}, \Pi_U )$is an analytic covering of$\mathcal{U}$(we say that$\mathcal{U}$has the$( \Pi_U, \mathcal{C} )$-\textit{covering property}). This guarantees that the cardinal of$\Pi_U^{- 1} ( \mathcal{u} ) \cap \mathcal{C}$is locally constant on$\mathcal{U}$and that$\Pi_U^{- 1} ( \mathcal{U} ) \cap \mathcal{C}$is a finite collection of sheets which are all locally homeomorphic to$\mathcal{U}$. In the case where$\Pi_U ( \mathcal{C} )$is dense in$\C all the known algorithms for solving $\mathcal{C}$ or $\mathcal{S}$ compute implicitly or explicitly a Zariski closed subset $W$ such that any sub-manifold of \Csetminus W have the ($\Pi_U, \mathcal{C}$)-covering property. We introduce the \textit{discriminant varieties of $\mathcal{C}$ w.r.t. $\Pi_U$} which are algebraic sets with the above property (even in the cases where $\Pi_U$ is not dense in \C. We then show that the set of points of \overline{\Pi_U ( \mathcal{C} )}$which do not have any neighborhood with the ($\Pi_U,\mathcal{C})-covering property is a Zariski closed set and thus the \textit{minimal discriminant variety of \mathcal{C}$w.r.t.$\Pi_U} and we propose an algorithm to compute it efficiently. Thus, solving the parametric system \mathcal{C}$(resp.$\mathcal{S}$) then remains to describe$\Csetminus W_D$(resp.$\R^d \setminus W_D$) which can be done using critical points method or partial CAD based strategies. We did not fully study the complexity, but in the case of systems where$\overline{\Pi_U ( \mathcal{C} )} = \C the degree of the minimal discriminant variety as well as the running time of an algorithm able to compute it are singly exponential in the number of variables according to already known results

In Memory of Vladimir Gerdt

Center for Computational Methods in Applied Mathematics of RUDN, Professor V.P. Gerdt, whose passing was a great loss to the scientific center and the computer algebra community. The article provides biographical information about V.P. Gerdt, talks about his contribution to the development of computer algebra in Russia and the world. At the end there are the author’s personal memories of V.P. Gerdt.Настоящая статья - мемориальная, она посвящена памяти руководителя научного центра вычислительных методов в прикладной математике РУДН, профессора В.П. Гердта, чей уход стал невосполнимой потерей для научного центра и всего сообщества компьютерной алгебры. В статье приведены биографические сведения о В.П. Гердте, рассказано о его вкладе в развитие компьютерной алгебры в России и мире. В конце приведены личные воспоминания автора о В.П. Гердте