We present a new technique, based on polynomial continuation, for solving sys- tems of n polynomi... more We present a new technique, based on polynomial continuation, for solving sys- tems of n polynomials in N complex variables. The method allows equations to be introduced one-by-one or in groups, obtaining at each stage a represen- tation of the solution set that can be extended to the next stage until nally obtaining the solution set for the entire system.
This article presents several numerical algorithms for computa- tions in sheaf cohomology. Let X ... more This article presents several numerical algorithms for computa- tions in sheaf cohomology. Let X be an algebraic set deflned by a system of homogeneous multivariate polynomials with coe-cients in C. Let C be a union of reduced, irreducible pure-dimensional curve components of X. The flrst algorithm computes the dimension of the flrst cohomology of any twist of the ideal sheaf
Path tracking is the fundamental computational tool in homotopy con- tinuation and is therefore k... more Path tracking is the fundamental computational tool in homotopy con- tinuation and is therefore key in most algorithms in the emerging eld of numerical algebraic geometry. Though the basic notions of predictor-corrector methods have been known for years, there is still much to be considered, particularly in the specialized algebraic setting of solving polynomial systems. This article considers two ways
This article presents several numerical algorithms for computations in sheaf cohomology. Let X be... more This article presents several numerical algorithms for computations in sheaf cohomology. Let X be an algebraic set defined by a system of homogeneous multivariate polynomials with coefficients in C. Let C be a union of reduced, irreducible pure-dimensional curve components of X. The first algorithm computes the dimension of the first cohomology of any twist of the ideal sheaf of C. Let D be a reduced set of points on C. The second and third algorithms solve the Riemann-Roch problem of computing the dimension of the space of divisors on C which are linearly equivalent to D. Let S be a reduced, connected, locally Cohen-Macaulay pure-dimensional surface made up of components of X. The fourth algorithm computes the first and second cohomology of any twist of the ideal sheaf of S. Furthermore, as the algorithms are based on homotopy continuation, they take advantage of the natural parallelism underlying continuation methods.
This paper illustrates how methods such as homotopy continuation and monodromy, when combined wit... more This paper illustrates how methods such as homotopy continuation and monodromy, when combined with a numerical version of Terracini's lemma, can be used to produce a high probability algorithm for com-puting the dimensions of secant and join varieties. The use of numerical methods allows applications to problems that are difficult to handle by purely symbolic algorithms.
47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, 2009
One-dimensional slow invariant manifolds for dynamical systems arising from modeling unsteady, is... more One-dimensional slow invariant manifolds for dynamical systems arising from modeling unsteady, isothermal, isochoric, spatially homogeneous, closed reactive systems are calculated. The technique is based on global analysis of the composition space of the reactive system. The identification of all the system's finite and infinite critical points plays a major role in calculating the system's slow invariant manifold. The slow invariant manifolds are constructed by calculating heteroclinic orbits which connect appropriate critical points to the critical point which corresponds to the unique stable physical critical point of chemical equilibrium. The technique is applied to small and large detailed kinetics mechanisms for hydrogen combustion.
Discrete and Continuous Dynamical Systems - Series S, 2010
The reaction-diffusion system modeling the dorsal-ventral patterning during the zebrafish embryo ... more The reaction-diffusion system modeling the dorsal-ventral patterning during the zebrafish embryo development, developed in [Y.-T. Zhang, A.D. Lander, Q. Nie, Journal of Theoretical Biology, 248 (2007), 579-589] has multiple steady state solutions. In this paper, we describe the computation of seven steady state solutions found by discretizing the boundary value problem using a finite difference scheme and solving the resulting polynomial system using algorithms from numerical algebraic geometry. The stability of each of these steady state solutions is studied by mathematical analysis and numerical simulations via a time marching approach. The results of this paper show that three of the seven steady state solutions are stable and the location of the organizer of a zebrafish embryo determines which stable steady state pattern the multi-stability system converges to. Numerical simulations also show that the system is robust with respect to the change of the organizer size.
The IMA Volumes in Mathematics and its Applications, 2008
By a numerical continuation method called a diagonal homotopy, one can compute the intersection o... more By a numerical continuation method called a diagonal homotopy, one can compute the intersection of two irreducible positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure to intersect general solution sets that are not necessarily irreducible or even equidimensional. Of particular interest is the special case where one of the sets is defined by a single polynomial equation. This leads to an algorithm for finding a numerical representation of the solution set of a system of polynomial equations introducing the equations one by one. Preliminary computational experiments show this approach can exploit the special structure of a polynomial system, which improves the performance of the path following algorithms.
One can consider the problem of factoring multivariate complex polynomials as a special case of t... more One can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components. The importance and nature of this problem however justify a special treatment. We exploit the reduction to the univariate root finding problem as a way to sample the polynomial more efficiently, certify the decomposition with linear traces, and apply interpolation techniques to construct the irreducible factors. With a random combination of differentials we lower multiplicities and reduce to the regular case. Estimates on the location of the zeroes of the derivative of polynomials provide bounds on the required precision. We apply our software to study the singularities of Stewart-Gough platforms.
Homotopy continuation provides a numerical tool for computing the equivalence of a smooth variety... more Homotopy continuation provides a numerical tool for computing the equivalence of a smooth variety in an intersection product. Intersection theory provides a theoretical tool for relating the equivalence of a smooth variety in an intersection product to the degrees of the Chern classes of the variety. A combination of these tools leads to a numerical method for computing the degrees of Chern classes of smooth projective varieties in P n . We illustrate the approach through several worked examples.
Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety , where... more Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety , where L 1 , . . . , L ρ is a basis of Pic(X ), ρ being the Picard number of X , and x 1 , . . . , x ρ are complex variables. After studying general properties of V X we specialize to the Hilbert curve of a polarized variety (X, L), namely the plane curve of degree dim(X ) associated to χ (xK X + yL). Special emphasis is given to the case of polarized threefolds.
Numerical di-culties encountered when following paths using methods such as homotopy continuation... more Numerical di-culties encountered when following paths using methods such as homotopy continuation may be overcome by combining adaptive stepsize and adaptive multiprecision. In the paper Adaptive multipreci- sion path tracking (1), precision and stepsize are adapted separately. This can lead to suboptimal performance and even failure in certain circum- stances. This paper presents a strategy for adjusting precision and stepsize
We present a new technique, based on polynomial continuation, for solving sys- tems of n polynomi... more We present a new technique, based on polynomial continuation, for solving sys- tems of n polynomials in N complex variables. The method allows equations to be introduced one-by-one or in groups, obtaining at each stage a represen- tation of the solution set that can be extended to the next stage until nally obtaining the solution set for the entire system.
This article presents several numerical algorithms for computa- tions in sheaf cohomology. Let X ... more This article presents several numerical algorithms for computa- tions in sheaf cohomology. Let X be an algebraic set deflned by a system of homogeneous multivariate polynomials with coe-cients in C. Let C be a union of reduced, irreducible pure-dimensional curve components of X. The flrst algorithm computes the dimension of the flrst cohomology of any twist of the ideal sheaf
Path tracking is the fundamental computational tool in homotopy con- tinuation and is therefore k... more Path tracking is the fundamental computational tool in homotopy con- tinuation and is therefore key in most algorithms in the emerging eld of numerical algebraic geometry. Though the basic notions of predictor-corrector methods have been known for years, there is still much to be considered, particularly in the specialized algebraic setting of solving polynomial systems. This article considers two ways
This article presents several numerical algorithms for computations in sheaf cohomology. Let X be... more This article presents several numerical algorithms for computations in sheaf cohomology. Let X be an algebraic set defined by a system of homogeneous multivariate polynomials with coefficients in C. Let C be a union of reduced, irreducible pure-dimensional curve components of X. The first algorithm computes the dimension of the first cohomology of any twist of the ideal sheaf of C. Let D be a reduced set of points on C. The second and third algorithms solve the Riemann-Roch problem of computing the dimension of the space of divisors on C which are linearly equivalent to D. Let S be a reduced, connected, locally Cohen-Macaulay pure-dimensional surface made up of components of X. The fourth algorithm computes the first and second cohomology of any twist of the ideal sheaf of S. Furthermore, as the algorithms are based on homotopy continuation, they take advantage of the natural parallelism underlying continuation methods.
This paper illustrates how methods such as homotopy continuation and monodromy, when combined wit... more This paper illustrates how methods such as homotopy continuation and monodromy, when combined with a numerical version of Terracini's lemma, can be used to produce a high probability algorithm for com-puting the dimensions of secant and join varieties. The use of numerical methods allows applications to problems that are difficult to handle by purely symbolic algorithms.
47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, 2009
One-dimensional slow invariant manifolds for dynamical systems arising from modeling unsteady, is... more One-dimensional slow invariant manifolds for dynamical systems arising from modeling unsteady, isothermal, isochoric, spatially homogeneous, closed reactive systems are calculated. The technique is based on global analysis of the composition space of the reactive system. The identification of all the system's finite and infinite critical points plays a major role in calculating the system's slow invariant manifold. The slow invariant manifolds are constructed by calculating heteroclinic orbits which connect appropriate critical points to the critical point which corresponds to the unique stable physical critical point of chemical equilibrium. The technique is applied to small and large detailed kinetics mechanisms for hydrogen combustion.
Discrete and Continuous Dynamical Systems - Series S, 2010
The reaction-diffusion system modeling the dorsal-ventral patterning during the zebrafish embryo ... more The reaction-diffusion system modeling the dorsal-ventral patterning during the zebrafish embryo development, developed in [Y.-T. Zhang, A.D. Lander, Q. Nie, Journal of Theoretical Biology, 248 (2007), 579-589] has multiple steady state solutions. In this paper, we describe the computation of seven steady state solutions found by discretizing the boundary value problem using a finite difference scheme and solving the resulting polynomial system using algorithms from numerical algebraic geometry. The stability of each of these steady state solutions is studied by mathematical analysis and numerical simulations via a time marching approach. The results of this paper show that three of the seven steady state solutions are stable and the location of the organizer of a zebrafish embryo determines which stable steady state pattern the multi-stability system converges to. Numerical simulations also show that the system is robust with respect to the change of the organizer size.
The IMA Volumes in Mathematics and its Applications, 2008
By a numerical continuation method called a diagonal homotopy, one can compute the intersection o... more By a numerical continuation method called a diagonal homotopy, one can compute the intersection of two irreducible positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure to intersect general solution sets that are not necessarily irreducible or even equidimensional. Of particular interest is the special case where one of the sets is defined by a single polynomial equation. This leads to an algorithm for finding a numerical representation of the solution set of a system of polynomial equations introducing the equations one by one. Preliminary computational experiments show this approach can exploit the special structure of a polynomial system, which improves the performance of the path following algorithms.
One can consider the problem of factoring multivariate complex polynomials as a special case of t... more One can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components. The importance and nature of this problem however justify a special treatment. We exploit the reduction to the univariate root finding problem as a way to sample the polynomial more efficiently, certify the decomposition with linear traces, and apply interpolation techniques to construct the irreducible factors. With a random combination of differentials we lower multiplicities and reduce to the regular case. Estimates on the location of the zeroes of the derivative of polynomials provide bounds on the required precision. We apply our software to study the singularities of Stewart-Gough platforms.
Homotopy continuation provides a numerical tool for computing the equivalence of a smooth variety... more Homotopy continuation provides a numerical tool for computing the equivalence of a smooth variety in an intersection product. Intersection theory provides a theoretical tool for relating the equivalence of a smooth variety in an intersection product to the degrees of the Chern classes of the variety. A combination of these tools leads to a numerical method for computing the degrees of Chern classes of smooth projective varieties in P n . We illustrate the approach through several worked examples.
Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety , where... more Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety , where L 1 , . . . , L ρ is a basis of Pic(X ), ρ being the Picard number of X , and x 1 , . . . , x ρ are complex variables. After studying general properties of V X we specialize to the Hilbert curve of a polarized variety (X, L), namely the plane curve of degree dim(X ) associated to χ (xK X + yL). Special emphasis is given to the case of polarized threefolds.
Numerical di-culties encountered when following paths using methods such as homotopy continuation... more Numerical di-culties encountered when following paths using methods such as homotopy continuation may be overcome by combining adaptive stepsize and adaptive multiprecision. In the paper Adaptive multipreci- sion path tracking (1), precision and stepsize are adapted separately. This can lead to suboptimal performance and even failure in certain circum- stances. This paper presents a strategy for adjusting precision and stepsize
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