On the Complexity of Computing Real Radicals of Polynomial Systems

Din, Mohab Safey El; Yang, Zhihong; Zhi, Lihong

doi:10.1145/3208976.3209002

Cited by 10 publications

(5 citation statements)

References 38 publications

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“…A more modern approach, introduced in [11] and refined in [3] allows one to compute sample points per connected components of basic semialgebraic sets in time which is singly exponential in the number of variables and polynomial in the number of input polynomials and their maximum degree. We refer the reader to [4] for the foundations of such approaches and to [23] for more modern algorithms based on the critical point method and to the Maple package RAGlib [22] which implements them. The two main functions which are provided by this Maple package RAGlib are HasRealSolutions and PointsPerComponents.…”

Section: Reduction To Polynomial System Solvingmentioning

confidence: 99%

Stability analysis of a bacterial growth model through computer algebra

Yabo,

Safey El Din,

Caillau

et al. 2023

MathematicS in Action

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We describe microbial growth and production of value-added chemical compounds in a continuous bioreactor through a dynamical system and we study the local stability of the equilibrium of interest by means of the classical Routh-Hurwitz criterion. The mathematical model considers various biological and structural parameters related to the bioprocess (concentration of substrate inflow, constants of the microchemical reactions, steady-state mass fractions of intracellular proteins, etc.) and thus, the stability condition is given in terms of these parameters. This boils down to deciding the consistency of a system of polynomial inequalities over the reals, which is challenging to solve from an analytical perspective, and out of reach even for traditional computational software designed to solve such problems. We show how to adapt classical techniques for solving polynomial systems to cope with this problem within a few minutes by leveraging its structural properties, thus completing the stability analysis of our model. The paper is accompanied by a Maple worksheet available online.

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Section: Reduction To Polynomial System Solvingmentioning

confidence: 99%

Stability analysis of a bacterial growth model through computer algebra

Yabo,

Safey El Din,

Caillau

et al. 2023

MathematicS in Action

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“…Complexity analysis in Real Algebraic Geometry is an active area of research, where obtaining good upper bounds is challenging. See for instance [LPR20] for elementary recursive degree bounds in Kivrine-Stengle Positivstellensatz, [SEDYZ18] for computation complexity of real radicals. Among all the Stellensätzen, we consider Putinar's Positivstellensatz, which allows a denominator free representation and has well-know applications in Polynomial Optimization.…”

Section: Related Workmentioning

confidence: 99%

On the Effective Putinar's Positivstellensatz and Moment Approximation

Baldi¹,

Mourrain²

2021

Preprint

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We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar Positivstellensatz over a compact basic closed semialgebraic set S, with new polynomial bounds on the degree of the positivity certificates. These bounds involve a Łojasiewicz exponent associated to the description of S. We show that under Constraint Qualification Conditions, this Łojasiewicz exponent is equal to 1. We deduce new bounds on the convergence rate of the optima in Lasserre Sum-of-Squares hierarchy to the global optimum of a polynomial function on S and new bounds on the Hausdorff distance between the cone of truncated (probability) measures supported on S and the cone of truncated moment sequences, which are positive on the quadratic module of S.

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“…Current algorithms and implementations of partial CAD and single open cell CAD can be found in [15,16,17]. Alternatively, the best current symbolic bound using quantifier elimination is provided by Basu, Pollack, and Roy [9] and the most recent implementation utilizing it can be found in [53].…”

Section: Related Workmentioning

confidence: 99%

“…Decompositions of semi-algebraic sest into a union of regular semi-algebraic sets using triangular decompositions, border bases, and moment matrices are detailed in [19,21,20]. The most recent implementation can be found in [53]. This approach, however, computes iteratively singularities of singularities, which can increase the complexity significantly in the worst case.…”

Section: Related Workmentioning

confidence: 99%

Smooth Points on Semi-algebraic Sets

Harris¹,

Hauenstein²,

Szántó³

2020

Preprint

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Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. Existing methods to compute smooth points on semialgebraic sets use symbolic quantifier elimination tools. In this paper, we present a simple algorithm based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected compact component of a real (semi)-algebraic set. Our technique is intuitive in principal, performs well on previously difficult examples, and is straightforward to implement using existing numerical algebraic geometry software. The practical efficiency of our approach is demonstrated by solving a conjecture on the number of equilibria of the Kuramoto model for the n = 4 case. We also apply our method to design an efficient algorithm to compute the real dimension of (semi)-algebraic sets, the original motivation for this research.

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On the Complexity of Computing Real Radicals of Polynomial Systems

Cited by 10 publications

References 38 publications

Stability analysis of a bacterial growth model through computer algebra

Stability analysis of a bacterial growth model through computer algebra

On the Effective Putinar's Positivstellensatz and Moment Approximation

Smooth Points on Semi-algebraic Sets

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