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Fabien Priziac

Publications and source records attributed to Fabien Priziac.

10 recordsLinked to original sources
arXiv2026

Quotients of real algebraic sets and AS-sets equipped with a linearizable compact group action

We present the full-detailed construction of new geometric quotient structures for affine real algebraic varieties equipped with a linearizable action of a compact Lie group G. These quotients are functors from the category of arc-symmetric/AS-sets equipped with a linearizable action of G and equivariant continuous maps with AS-graph to the category of AS-sets. We furthermore provide a complete review of the results employed in the constructions, including properties of equivariant real algebraic geometry with respect to polynomial group actions, as well as an introduction to semialgebraic arc-symmetric sets and AS-sets of compact affine Nash manifolds.

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arXiv2022

Distance to a constitutive tensor isotropy stratum by Lasserre polynomial optimization method

We give a detailed description of a polynomial optimization method allowing to solve a problem in continuum mechanics: the determination of the elasticity or the piezoelectricity tensor of a specific isotropy stratum the closest to a given experimental tensor, and the calculation of the distance to the given tensor from the considered isotropy stratum. We take advantage of the fact that the isotropy strata are semialgebraic sets to show that the method, developed by Lasserre and coworkers which consists in solving polynomial optimization problems with semialgebraic constraints, successfully applies.

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arXiv2022

The distance to cubic symmetry class as a polynomial optimization problem

Generically, a fully measured elasticity tensor has no material symmetry. For single crystals with a cubic lattice, or for the aeronautics turbine blades superalloys such as Nickelbased CMSX-4, cubic symmetry is nevertheless expected. It is in practice necessary to compute the nearest cubic elasticity tensor to a given raw one. Mathematically formulated, the problem consists in finding the distance between a given tensor and the cubic symmetry stratum. It is known that closed symmetry strata (for any tensorial representation of the rotation group) are semialgebraic sets, defined by polynomial equations and inequalities. It has been recently shown that the closed cubic elasticity stratum is moreover algebraic, which means that it can be defined by polynomial equations only (without requirement to polynomial inequalities). We propose to make use of this mathematical property to formulate the distance to cubic symmetry problem as a polynomial (in fact quadratic) optimization problem, and to derive its quasi-analytical solution using the technique of Gr{ö}bner bases. The proposed methodology also applies to cubic Hill elasto-plasticity (where two fourth-order constitutive tensors are involved).

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arXiv2017

Products of real equivariant weight filtrations

We first show the existence of a weight filtration on the equivariant cohomology of real algebraic varieties equipped with the action of a finite group, by applying group cohomology to the dual geometric filtration. We then prove the compatibility of the equivariant weight filtrations and spectral sequences with K{ü}nneth isomorphism, cup and cap products, from the filtered chain level. We finally induce the usual formulae for the equivariant cup and cap products from their analogs on the non-equivariant weight spectral sequences.

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arXiv2017

Equivariant weight filtration for real algebraic varieties with action

We show the existence of an equivariant weight filtration on the equivariant homology of real algebraic varieties equipped with a finite group action, by applying group homology onto the weight complex of McCrory and Parusiński. The group action enriches the information contained in the induced equivariant weight spectral sequence from which we can not extract finite additivity directly if the group is of even order. Still, we manage to construct additive invariants from its elementary bricks, and prove that they are induced from the very equivariant geometry of real algebraic varieties with action, and that they coincide with Fichou's equivariant virtual Betti numbers in some cases. In the case of the two-elements group, we recover these through the use of globally invariant chains and the equivariant version of Guill{é}n and Navarro Aznar's extension criterion.

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arXiv2016

On the equivariant blow-Nash classification of simple invariant Nash germs

We make progress towards the classification of simple Nash germs invariant under the involution changing the sign of the first coordinate, with respect to equivariant blow-Nash equivalence, which is an equivariant Nash version of blow-analytic equivalence, taking advantage of invariants for this relation, the equivariant zeta functions.

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arXiv2016

Equivariant zeta functions for invariant Nash germs

To any Nash germ invariant under right composition with a linear action of a finite group, we associate its equivariant zeta functions, inspired from motivic zeta functions, using the equivariant virtual Poincaré series as a motivic measure. We show Denef-Loeser formulae for the equivariant zeta functions and prove that they are invariants for equivariant blow-Nash equivalence via equivariant blow-Nash isomorphisms. Equivariant blow-Nash equivalence between invariant Nash germs is defined as a generalization involving equivariant data of the blow-Nash equivalence.

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arXiv2016

Cohomology and products of real weight filtrations

We associate to each algebraic variety defined over $\mathbb{R}$ a filtered cochain complex, which computes the cohomology with compact supports and $\mathbb{Z}\_2$-coefficients of the set of its real points. This filtered complex is additive for closed inclusions and acyclic for resolution of singularities, and is unique up to filtered quasi-isomorphism. It is represented by the dual filtration of the geometric filtration on semialgebraic chains with closed supports defined by McCrory and Parusiński, and leads to a spectral sequence which computes the weight filtration on cohomology with compact supports. This spectral sequence is a natural invariant which contains the additive virtual Betti numbers. We then show that the cross product of two varieties allows us to compare the product of their respective weight complexes and spectral sequences with those of their product, and prove that the cup and cap products in cohomology and homology are filtered with respect to the real weight filtrations.

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arXiv2013

Complexe de poids des variétés algébriques réelles avec action

Using the functoriality of C. McCrory and A. Parusiński's weight complex -which induces an analog of the weight filtration for complex algebraic varieties on the Borel-Moore homology with $\mathbb{Z}_2$ coefficients of real algebraic varieties-, we define a weight complex with action on the real algebraic varieties equipped with a finite group action. Emphasizing on the two elements group, we then establish a filtered version of Smith short sequence, taking into account the Nash-constructible filtration which realizes the weight complex with action. Its exactness is implied by the splitting of a Nash manifold equipped with an algebraic involution along an arc-symmetric subset.

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