3.1 High order geometric modeling

The accurate description of shapes is a long standing problem in mathematics, with an important impact in many domains, inducing strong interactions between geometry and computation. Developing precise geometric modeling techniques is a critical issue in CAD-CAM. Constructing accurate models, that can be exploited in geometric applications, from digital data produced by cameras, laser scanners, observations or simulations is also a major issue in geometry processing. A main challenge is to construct models that can capture the geometry of complex shapes, using few parameters while being precise.

Our first objective is to develop methods, which are able to describe accurately and in an efficient way, objects or phenomena of geometric nature, using algebraic representations.

The approach followed in CAGD, to describe complex geometry is based on parametric representations called NURBS (Non Uniform Rational B-Spline). The models are constructed by trimming and gluing together high order patches of algebraic surfaces. These models are built from the so-called B-Spline functions that encode a piecewise algebraic function with a prescribed regularity at knots. Although these models have many advantages and have become the standard for designing nowadays CAD models, they also have important drawbacks. Among them, the difficulty to locally refine a NURBS surface and also the topological rigidity of NURBS patches that imposes to use many such patches with trims for designing complex models, with the consequence of the appearing of cracks at the seams. To overcome these difficulties, an active area of research is to look for new blending functions for the representation of CAD models. Some examples are the so-called T-Splines, LR-Spline blending functions, or hierarchical splines, that have been recently devised in order to perform efficiently local refinement. An important problem is to analyze spline spaces associated to general subdivisions, which is of particular interest in higher order Finite Element Methods. Another challenge in geometric modeling is the efficient representation and/or reconstruction of complex objects, and the description of computational domains in numerical simulation. To construct models that can represent efficiently the geometry of complex shapes, we are interested in developing modeling methods, based on alternative constructions such as skeleton-based representations. The change of representation, in particular between parametric and implicit representations, is of particular interest in geometric computations and in its applications in CAGD.

We also plan to investigate adaptive hierarchical techniques, which can locally improve the approximation of a shape or a function. They shall be exploited to transform digital data produced by cameras, laser scanners, observations or simulations into accurate and structured algebraic models.

The precise and efficient representation of shapes also leads to the problem of extracting and exploiting characteristic properties of shapes such as symmetry, which is very frequent in geometry. Reflecting the symmetry of the intended shape in the representation appears as a natural requirement for visual quality, but also as a possible source of sparsity of the representation. Recognizing, encoding and exploiting symmetry requires new paradigms of representation and further algebraic developments. Algebraic foundations for the exploitation of symmetry in the context of non linear differential and polynomial equations are addressed. The intent is to bring this expertise with symmetry to the geometric models and computations developed by aromath.

3.2 Robust algebraic-geometric computation

In many problems, digital data are approximated and cannot just be used as if they were exact. In the context of geometric modeling, polynomial equations appear naturally as a way to describe constraints between the unknown variables of a problem. An important challenge is to take into account the input error in order to develop robust methods for solving these algebraic constraints. Robustness means that a small perturbation of the input should produce a controlled variation of the output, that is forward stability, when the input-output map is regular. In non-regular cases, robustness also means that the output is an exact solution, or the most coherent solution, of a problem with input data in a given neighborhood, that is backward stability.

Our second long term objective is to develop methods to robustly and efficiently solve algebraic problems that occur in geometric modeling.

Robustness is a major issue in geometric modeling and algebraic computation. Classical methods in computer algebra, based on the paradigm of exact computation, cannot be applied directly in this context. They are not designed for stability against input perturbations. New investigations are needed to develop methods which integrate this additional dimension of the problem. Several approaches are investigated to tackle these difficulties.

One relies on linearization of algebraic problems based on “elimination of variables” or projection into a space of smaller dimension. Resultant theory provides a strong foundation for these methods, connecting the geometric properties of the solutions with explicit linear algebra on polynomial vector spaces, for families of polynomial systems (e.g., homogeneous, multi-homogeneous, sparse). Important progress has been made in the last two decades to extend this theory to new families of problems with specific geometric properties. Additional advances have been achieved more recently to exploit the syzygies between the input equations. This approach provides matrix based representations, which are particularly powerful for approximate geometric computation on parametrized curves and surfaces. They are tuned to certain classes of problems and an important issue is to detect and analyze degeneracies and to adapt them to these cases.

A more adaptive approach involves linear algebra computation in a hierarchy of polynomial vector spaces. It produces a description of quotient algebra structures, from which the solutions of polynomial systems can be recovered. This family of methods includes Gröbner Basis , which provides general tools for solving polynomial equations. Border Basis is an alternative approach, offering numerically stable methods for solving polynomial equations with approximate coefficients . An important issue is to understand and control the numerical behavior of these methods as well as their complexity and to exploit the structure of the input system.

In order to compute “only” the (real) solutions of a polynomial system in a given domain, duality techniques can also be employed. They consist in analyzing and adding constraints on the space of linear forms which vanish on the polynomial equations. Combined with semi-definite programming techniques, they provide efficient methods to compute the real solutions of algebraic equations or to solve polynomial optimization problems. The main issues are the completness of the approach, their scalability with the degree and dimension and the certification of bounds.

Singular solutions of polynomial systems can be analyzed by computing differentials, which vanish at these points. This leads to efficient deflation techniques, which transform a singular solution of a given problem into a regular solution of the transformed problem. These local methods need to be combined with more global root localisation methods.

Subdivision methods are another type of methods which are interesting for robust geometric computation. They are based on exclusion tests which certify that no solution exists in a domain and inclusion tests, which certify the uniqueness of a solution in a domain. They have shown their strength in addressing many algebraic problems, such as isolating real roots of polynomial equations or computing the topology of algebraic curves and surfaces. The main issues in these approaches is to deal with singularities and degenerate solutions.

4.1 Geometric modeling for Design and Manufacturing.

The main domain of applications that we consider for the methods we develop is Computer Aided Design and Manufacturing.

Computer-Aided Design (CAD) involves creating digital models defined by mathematical constructions, from geometric, functional or aesthetic considerations. Computer-aided manufacturing (CAM) uses the geometrical design data to control the tools and processes, which lead to the production of real objects from their numerical descriptions.

CAD-CAM systems provide tools for visualizing, understanding, manipulating, and editing virtual shapes. They are extensively used in many applications, including automotive, shipbuilding, aerospace industries, industrial and architectural design, prosthetics, and many more. They are also widely used to produce computer animation for special effects in movies, advertising and technical manuals, or for digital content creation. Their economic importance is enormous. Their importance in education is also growing, as they are more and more used in schools and educational purposes.

CAD-CAM has been a major driving force for research developments in geometric modeling, which leads to very large software, produced and sold by big companies, capable of assisting engineers in all the steps from design to manufacturing.

Nevertheless, many challenges still need to be addressed. Many problems remain open, related to the use of efficient shape representations, of geometric models specific to some application domains, such as in architecture, naval engineering, mechanical constructions, manufacturing ...Important questions on the robustness and the certification of geometric computation are not yet answered. The complexity of the models which are used nowadays also appeals for the development of new approaches. The manufacturing environment is also increasingly complex, with new type of machine tools including: turning, 5-axes machining and wire EDM (Electrical Discharge Machining), 3D printer. It cannot be properly used without computer assistance, which raises methodological and algorithmic questions. There is an increasing need to combine design and simulation, for analyzing the physical behavior of a model and for optimal design.

The field has deeply changed over the last decades, with the emergence of new geometric modeling tools built on dedicated packages, which are mixing different scientific areas to address specific applications. It is providing new opportunities to apply new geometric modeling methods, output from research activities.

4.2 Geometric modeling for Numerical Simulation and Optimization

A major bottleneck in the CAD-CAM developments is the lack of interoperability of modeling systems and simulation systems. This is strongly influenced by their development history, as they have been following different paths.

The geometric tools have evolved from supporting a limited number of tasks at separate stages in product development and manufacturing, to being essential in all phases from initial design through manufacturing.

Current Finite Element Analysis (FEA) technology was already well established 40 years ago, when CAD-systems just started to appear, and its success stems from using approximations of both the geometry and the analysis model with low order finite elements (most often of degree $\le 2$).

There has been no requirement between CAD and numerical simulation, based on Finite Element Analysis, leading to incompatible mathematical representations in CAD and FEA. This incompatibility makes interoperability of CAD/CAM and FEA very challenging. In the general case today this challenge is addressed by expensive and time-consuming human intervention and software developments.

Improving this interaction by using adequate geometric and functional descriptions should boost the interaction between numerical analysis and geometric modeling, with important implications in shape optimization. In particular, it could provide a better feedback of numerical simulations on the geometric model in a design optimization loop, which incorporates iterative analysis steps.

The situation is evolving. In the past decade, a new paradigm has emerged to replace the traditional Finite Elements by B-Spline basis element of any polynomial degree, thus in principle enabling exact representation of all shapes that can be modeled in CAD. It has been demonstrated that the so-called isogeometric analysis approach can be far more accurate than traditional FEA.

It opens new perspectives for the interoperability between geometric modeling and numerical simulation. The development of numerical methods of high order using a precise description of the shapes raises questions on piecewise polynomial elements, on the description of computational domains and of their interfaces, on the construction of good function spaces to approximate physical solutions. All these problems involve geometric considerations and are closely related to the theory of splines and to the geometric methods we are investigating. We plan to apply our work to the development of new interactions between geometric modeling and numerical solvers.

6.1 Elimination ideals and Bézout relations

Participant: André Galligo.

This is a joint work with Zbigniew Jelonek, Instytut Matematyczny PAN, Warszawa, Poland.

6.2 Separation bounds for polynomial systems

Participant: Ioannis Emiris, Bernard Mourrain, Elias Tsigaridas (OURAGAN).

We rely on aggregate separation bounds for univariate polynomials to introduce, in 24, novel worst-case separation bounds for the isolated roots of zero-dimensional, positive-dimensional, and overdetermined polynomial systems. We exploit the structure of the given system, as well as bounds on the height of the sparse (or toric) resultant, by means of mixed volume, thus establishing adaptive bounds. Our bounds improve upon Canny’s Gap theorem. Moreover, they exploit sparseness and they apply without any assumptions on the input polynomial system. To evaluate the quality of the bounds, we present polynomial systems whose root separation is asymptotically not far from our bounds. We apply our bounds to three problems. First, we use them to estimate the bitsize of the eigenvalues and eigenvectors of an integer matrix; thus we provide a new proof that the problem has polynomial bit complexity. Second, we bound the value of a positive polynomial over the simplex: we improve by at least one order of magnitude upon all existing bounds. Finally, we asymptotically bound the number of steps of any purely subdivision-based algorithm that isolates all real roots of a polynomial system.

6.3 Matrix formulae for Resultants and Discriminants of Bivariate Tensor-product Polynomials

Participant: Laurent Busé, Angelos Mantzaflaris, Elias Tsigaridas (OURAGAN).

The construction of optimal resultant formulae for polynomial systems is one of the main areas of research in computational algebraic geometry. However, most of the constructions are restricted to formulae for unmixed polynomial systems, that is, systems of polynomials which all have the same support. Such a condition is restrictive, since mixed systems of equations arise frequently in many problems. Nevertheless, resultant formulae for mixed polynomial systems is a very challenging problem. In 21 we present a square, Koszul-type, matrix, the determinant of which is the resultant of an arbitrary (mixed) bivariate tensor-product polynomial system. The formula generalizes the classical Sylvester matrix of two univariate polynomials, since it expresses a map of degree one, that is, the elements of the corresponding matrix are up to sign the coefficients of the input polynomials. Interestingly, the matrix expresses a primal-dual multiplication map, that is, the tensor product of a univariate multiplication map with a map expressing derivation in a dual space. In addition we prove an impossibility result which states that for tensor-product systems with more than two (affine) variables there are no universal degree-one formulae, unless the system is unmixed. Last but not least, we present applications of the new construction in the efficient computation of discriminants and mixed discriminants.

6.4 Fibers of multi-graded rational maps and orthogonal projection onto rational surfaces

Participant: Laurent Busé, Fatmanur Yildirim.

In 16 we contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite fibers of a generically finite dominant rational map: a congruence of normal lines to the rational surface. Then, an in-depth study of certain syzygy modules associated to such a congruence is presented and applied to build elimination matrices that provide universal representations of its finite fibers, under some genericity assumptions. These matrices depend linearly in the variables of the three dimensional space. They can be pre-computed so that the orthogonal projections of points are approximately computed by means of fast and robust numerical linear algebra calculations.

This is a joint work with Nicolas Notbol, the University of Buenos Aires, Argentina and Marc Chardin, CNRS and Institut de Mathématiques de Jussieu, Sorbonne Universités, Paris, France.

6.5 Degree and birationality of multi-graded rational maps

Participant: Laurent Busé.

In 18 we give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian dual criterion to the multi-graded setting. Our approach is based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of the rational map. A key ingredient is a new algebra that we call the saturated special fiber ring, which turns out to be a fundamental tool to analyze the degree of a rational map. We also provide a very effective birationality criterion and a complete description of the equations of the associated Rees algebra of a particular class of plane rational maps.

This is a joint work Carlos D'Andrea and Yairon Cid Ruiz from the University of Barcelona, Spain.

6.6 Freeness and invariants of rational plane curves

Participant: Laurent Busé.

Given a parameterization φ of a rational plane curve C, in 20 we study some invariants of C via φ. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the pull-back of a line via φ, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via φ, we prove that the (nearly) freeness of a rational curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by product, we also show that the global Tjurina number of a rational curve can be computed directly from one of its parameterization, without relying on the computation of an equation of C.

This is a joint work Alexandru Dimca from the University Côté d'Azur, France, and Gabriel Sticlaru from the Faculty of Mathematics and Informatics of Constanta, Romania.

6.7 The geometry of the flex locus of a hypersurface

Participant: Laurent Busé.

In 19 we give a formula in terms of multidimensional resultants for an equation for the flex locus of a projective hypersurface, generalizing a classical result of Salmon for surfaces in P3. Using this formula, we compute the dimension of this flex locus, and an upper bound for the degree of its defining equations. We also show that, when the hypersurface is generic, this bound is reached, and that the generic flex line is unique and has the expected order of contact with the hypersurface.

This is a joint work Carlos D'Andrea and Martin Sombra from the University of Barcelona, Spain, and Martin Weimann from the University of Caen, France.

6.8 Regularity of bicyclic graphs and their powers

Participant: Navid Nemati.

In 22 we characterise the Castelnuovo-Mumford regularity of edge ideal $I\left(G\right)$ of a bicyclic graph $G$ in terms of the induced matching numbers of $G$. For the base case of this family of graphs, i.e. dumbbell graphs, we explicitly compute the induced matching number. Moreover, we prove that $\mathrm{reg}I{\left(G\right)}^q=2q+\mathrm{reg}I\left(G\right)-2$, for all $q\ge 1$, when $G$ is a dumbbell graph with a connecting path having no more than two vertices.

This is a joint work with Yairon Cid-Ruiz from Universitat de Barcelona, Spain, Sepehr Jafari from University Degli Studi di Genova, and Beatrice Picone from University of Catania, Italy.

6.9 Lefschetz properties of monomial algebras with almost linear resolution

Participant: Navid Nemati.

In 22 we study the WLP and SLP of artinian monomial ideals in $S=k[x_1,\cdots ,x_n]$ via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of $S/I$ is linear for at least $n-2$ steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial ideals with almost linear resolutions.

This is a joint work with Nasrin Altafi from KTH Royal Institute of Technology, Sweden.

6.10 Punctual Hilbert Schemes and Certified Approximate Singularities

Participant: Angelos Mantzaflaris, Bernard Mourrain.

In 35 we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system $f=(f_1,\cdots ,f_N)\in ℂ{[x_1,...,x_n]}^N$, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of $f$ such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so called inverse system that describes the multiplicity structure at the root. We use $\alpha$-theory test to certify the quadratic convergence, and to give bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.

This is a joint work with Agnes Szanto, Department of mathematics, North Carolina State University, USA.

6.11 Ideal Interpolation, H-Bases and Symmetry

Participant: Evelyne Hubert, Erick Rodriguez Bazan.

Multivariate Lagrange and Hermite interpolation are examples of ideal interpolation. The interpolation problem is defined by a set of linear forms, on the polynomial ring, whose kernels intersect into an ideal. Another class of examples thus arises when the linear forms are defined by the normal forms modulo an ideal. For an ideal interpolation problem with symmetry, we address in 36 the simultaneous computation of a symmetry adapted basis of the least interpolation space and the related symmetry adapted H-basis. The algorithm exploits symmetry at all stages, in addition to preserving it in the output. Defined by de Boor and Ron (1990), the least interpolation space is a canonically defined representation of the quotient of the polynomial ring by the ideal. As a result it has great properties that cannot be achieved by a space spanned by monomials. In particular it is invariant under a group action when the ideal is; Equivalently, when the set of linear forms is. Its natural complement is a H-basis of the ideal, as introduced by Macaulay (1916). As a contrast with Groebner bases, which are tied to a monomial term order and are notorious for not preserving symmetry, we show that symmetry can be naturally preserved and represented in H-bases, as well as exploited in the course of their computation.

6.12 Computing Free Non-commutative Groebner Bases over Z with Singular:Letterplace

Participant: Tobias Metzlaff.

This is joint work with Viktor Levandovskyy and Karim Zeid, RWTH Aachen University, Germany.

6.13 On minimal decompositions of low rank symmetric tensors

Participant: Bernard Mourrain.

This is a joint work with Alessandro Oneto, Univ. de Trento, Italy.

6.14 Computational geometry in general dimension

Participant: Ioannis Emiris, Ioannis Psarros.

Participant: Ioannis Emiris, Ioannis Psarros.

This is joint work with Zeta Avarikioti (ETH Zurich), and Loukas Kavouras (Athena RC).

Participant: Ioannis Emiris, Evangelos Bartzos.

This is joint work with Josef Schicho (U Linz).

6.15 Complete Classification and Efficient Determination of Arrangements Formed by Two Ellipsoids

Participant: Bernard Mourrain.

Arrangements of geometric objects refer to the spatial partitions formed by the objects and they serve as an underlining structure of motion design, analysis and planning in CAD/CAM, robotics, molecular modeling, manufacturing and computer-assisted radio-surgery. Arrangements are especially useful to collision detection, which is a key task in various applications such as computer animation , virtual reality, computer games, robotics, CAD/CAM and computational physics. Ellipsoids are commonly used as bounding volumes in approximating complex geometric objects in collision detection. In 28, we present an in-depth study on the arrangements formed by two ellipsoids. Specifically, we present a classification of these arrangements and propose an efficient algorithm for determining the arrangement formed by any particular pair of ellipsoids. A stratification diagram is also established to show the connections among all the arrangements formed by two ellipsoids. Our results for the first time elucidate all possible relative positions between two arbitrary ellipsoids and provides an efficient and robust algorithm for determining the relative position of any two given ellipsoids, therefore providing the necessary foundation for developing practical and trustworthy methods for processing ellipsoids for collision analysis or simulation in various applications.

This is a joint work with Xiaohong Jia, KLMM - Key Laboratory of Mathematics Mechanization, Beijing, China; Changhe Tu, School of Computer Science and Technology, Qingdao, China; Wenping Wang, HKU - The University of Hong Kong, China.

6.16 Interpolatory Catmull-Clark volumetric subdivision over unstructured hexahedral meshes for modeling and simulation applications

Participant: Bernard Mourrain.

Volumetric modeling is an important topic for material modeling and isogeometric simulation. In 33, two kinds of interpolatory Catmull-Clark volumetric subdivision approaches over unstructured hexahedral meshes are proposed based on the limit point formula of Catmull-Clark subdivision volume. The basic idea of the first method is to construct a new control lattice, whose limit volume by the Catmull-Clark subdivision scheme interpolates vertices of the original hexahedral mesh. The new control lattice is derived by the local push-back operation from one CatmullClark subdivision step with modified geometric rules. This interpolating method is simple and efficient, and several shape parameters are involved in adjusting the shape of the limit volume. The second method is based on progressive-iterative approximation using limit point formula. At each iteration step, we progressively modify vertices of an original hexahedral mesh to generate a new control lattice whose limit volume interpolates all vertices in the original hexahedral mesh. The convergence proof of the iterative process is also given. The interpolatory subdivision volume has C${}^2$-smoothness at the regular region except around extraordinary vertices and edges. Furthermore, the proposed interpolatory volumetric subdivision methods can be used not only for geometry interpolation, but also for material attribute interpolation in the field of volumetric material modeling. The application of the proposed volumetric subdivision approaches on isogeometric analysis is also given with several examples.

This is a joint work with Jin Xie, Jinlan Xu, Zhenyu Dong, Gang Xu, Chongyang Deng, HDU - College of Computer Science, Hangzhou, China; Yongjie Jessica Zhang, Department of Mechanics, Carnegie Mellon University, USA.

6.17 Geometrically smooth spline bases for data fitting and simulation

Participant: Ahmed Blidia, Bernard Mourrain.

This is joint work with Gang Xu, HDU - College of Computer Science, Hangzhou, China.

6.18 An overview of Geometry plus Simulation Modules

Participant: Angelos Mantzaflaris.

6.19 Local (T)HB-spline projectors via restricted hierarchical spline fitting

Participant: Angelos Mantzaflaris.

This is a joint work with Alessandro Giust and Bert Jüttler for the Johannes Kepler University of Linz, Austria.

6.20 Efficient Matrix Assembly in Isogeometric Analysis with Hierarchical B-splines

Participant: Angelos Mantzaflaris.

This is a joint work with Maodong Pan and Bert Jüttler for the Johannes Kepler University of Linz, Austria.

7.1 Bilateral contracts with industry

$•$ NURBSFIX: Repairing the topology of a NURBS model in view of its approximation. We have a research contract with the industrial partner GeometryFactory, in collaboration with the project-team Titane (Pierre Alliez). The post-doc of Xiao Xiao is funded by this research contract together with a PEPS from the labex AMIES. We continue to develop a robust algorithm which offers the capability to repair 155 while meshing CAD models, without requiring a valid surface mesh as input, and 156 to generate volume meshes which are valid by design and 157 independent of the input NURBS patch layout, without resorting 158 to post-processing steps such as mesh quilting or remeshing.

$•$ Geometric computing. Ioannis Emiris coordinates a research contract with the industrial partner ANSYS (Greece), in collaboration with Athena Research Center. PhD candidate A. Chalkis is partially funded.

Electronic design automation (EDA) and simulating Integrated Circuits requires robust geometric operations on thousands of electronic elements (capacitors, resistors, coils etc) represented by polyhedral objects in 2.5 dimensions, not necessarily convex. A special case may concern axis-aligned objects but the real challenge is the general case. The project, extended into 2021, focuses on 3 axes: (1) efficient data structures and prototype implementations for storing the aforementioned polyhedral objects so that nearest neighbor queries are fast in the L-max metric, which is the primary focus of the contract, (2) random sampling of the free space among objects, (3) data-driven algorithmic design for problems concerning data-structures and their construction and initialization.

7.2 Bilateral grants with industry

$•$ Interactive construction of 3D models - Application to the modeling of complex geological structures.

CIFRE collaboration between Schlumberger Montpellier (A. Azzedine) and Inria Sophia Antipolis (B. Mourrain). The PhD candidate is A. Belhachmi. The objective of the work is the development of a new spline based high quality geomodeler for reconstructing the stratigraphy of geological layers from the adaptive and efficient processing of large terrain information.

8.1 International initiatives

8.2 European initiatives

8.3 National initiatives

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

10.1 Major publications

• 1 articleEvangelosE. Bartzos, Ioannis Z.I. Emiris and JosefJ. Schicho. On the multihomogeneous Bézout bound on the number of embeddings of minimally rigid graphsApplicable Algebra in Engineering, Communication and Computing315-62020, 325-357
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• 2 articleLaurentL. Busé, YaironY. Cid-Ruiz and CarlosC. D'Andrea. Degree and birationality of multi-graded rational mapsProceedings of the London Mathematical Society12142020, 743-787
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• 3 articleLaurentL. Busé and AnnaA. Karasoulou. Resultant of an equivariant polynomial system with respect to the symmetric groupJournal of Symbolic Computation762016, 142-157
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• 4 articleIoannis Z.I. Emiris and IoannisI. Psarros. Products of Euclidean Metrics, Applied to Proximity Problems among CurvesACM Transactions on Spatial Algorithms and Systems64August 2020, 1-20
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• 5 articleAlvaro JavierA. Fuentes Suárez and EvelyneE. Hubert. Scaffolding skeletons using spherical Voronoi diagrams: feasibility, regularity and symmetryComputer-Aided Design102May 2018, 83 - 93
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• 6 articleAlessandroA. Giust, BertB. Jüttler and AngelosA. Mantzaflaris. Local (T)HB-spline projectors via restricted hierarchical spline fittingComputer Aided Geometric Design80June 2020, 101865
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• 7 articlePaulP. Görlach, EvelyneE. Hubert and ThéoT. Papadopoulo. Rational invariants of even ternary forms under the orthogonal groupFoundations of Computational Mathematics192019, 1315-1361
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• 8 articleZbigniewZ. Jelonek and AndréA. Galligo. Elimination ideals and Bezout relationsJournal of Algebra5622020, 621-626
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• 9 articleAngelosA. Mantzaflaris, BertB. Jüttler, BorisB. Khoromskij and UlrichU. Langer. Low Rank Tensor Methods in Galerkin-based Isogeometric AnalysisComputer Methods in Applied Mechanics and Engineering316April 2017, 1062-1085
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• 10 articleBernardB. Mourrain. Polynomial-Exponential Decomposition from MomentsFoundations of Computational Mathematics186December 2018, 1435--1492
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• 11 articleSimonS. Telen, BernardB. Mourrain and MarcM. Van Barel. Solving Polynomial Systems via a Stabilized Representation of Quotient AlgebrasSIAM Journal on Matrix Analysis and Applications393October 2018, 1421--1447
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10.2 Publications of the year

10.3 Cited publications