arXiv subjects

Hadi Seyedinejad

Publications and source records attributed to Hadi Seyedinejad.

8 recordsLinked to original sources

arXiv2018

Decomposition of sets in real algebraic geometry

We present a new notion of decomposition of semialgebraic sets by introducing a mode of irreducibility based on arc-analytic functions. The result is a refinement of the decomposition of such sets with respect to the Zariski topology as well as a refinement of the decomposition in each of the recent approaches based on Nash and continuous rational functions. In addition, by pairing the ring of arc-analytic functions with semialgebraic sets, we obtain a theory of algebraic geometry equipped with strong tools such as the Identity Principle and the Nullstellensatz.

math.AG↗

arXiv2018

On solutions of linear equations with polynomial coefficients

We show that a linear functional equation with polynomial coefficients need not admit an arc-analytic solution even if it admits a continuous semialgebraic one. We also show that such an equation need not admit a Nash regulous solution even if it admits an arc-analytic one.

math.AG↗

arXiv2018

Extensions of arc-analytic functions

We prove that every arc-analytic semialgebraic function on an arc-symmetric set admits an arc-analytic semialgebraic extension to the whole ambient Euclidean space.

math.AG↗

arXiv2017

A proof of Kurdyka's conjecture on arc-analytic functions

We prove a conjecture of Kurdyka stating that every arc-symmetric semialgebraic set is precisely the zero locus of an arc-analytic semialgebraic function. This implies, in particular, that arc-symmetric semialgebraic sets are in one-to-one correspondence with radical ideals of the ring of arc-analytic semialgebraic functions.

math.AG↗

arXiv2017

A fast flatness testing algorithm in characteristic zero

We prove a fast computable criterion that expresses non-flatness in terms of torsion: Let R be a regular algebra of finite type over a field K of characteristic zero and let F be a module finitely generated over an R-algebra of finite type. Given a maximal ideal m in R, let S be the coordinate ring of the blowing-up of Spec(R) at the closed point m. Then F is flat over R localized in m if and only if the tensor product of F with S over R is a torsion-free module over R localized in m. If K is the field of reals or complex numbers, we give a stronger criterion - without the regularity assumption on R. We also show the corresponding results in the real- and complex-analytic categories.

math.AC↗

arXiv2017

Flatness testing over singular bases

We show that non-flatness of a morphism f of complex-analytic spaces with a locally irreducible target Y of dimension n manifests in the existence of vertical components in the n-fold fibred power of the pull-back of f to the desingularization of Y. An algebraic analogue follows: Let R be a locally (analytically) irreducible finite type complex-algebra and an integral domain of Krull dimension n, and let S be a regular n-dimensional algebra of finite type over R (but not necessarily a finite R-module), such that the induced morphism of spectra is dominant. Then a finite type R-algebra A is R-flat if and only if the tensor product of S with the n-fold tensor power of A over R is a torsion-free R-module.

math.AC↗

arXiv2016

On topological classification of complex mappings

We study the topological invariant $ϕ$ of Kwieciński and Tworzewski, particularly beyond the case of mappings with smooth targets. We derive a lower bound for $ϕ$ of a general mapping, which is similarly effective as the upper bound given by Kwieciński and Tworzewski. Some classes of mappings are identified for which the exact value of $ϕ$ can be computed. Also, we prove that the variation of $ϕ$ on the source space of a mapping with a smooth target is semicontinuous in Zariski topology.

math.CV↗

arXiv2016

Finite determinacy and stability of flatness of analytic mappings

It is proved that flatness of an analytic mapping germ from a complete intersection is determined by its sufficiently high jet. As a consequence, one obtains finite determinacy of complete intersections. It is also shown that flatness and openness are stable under deformations.

math.CV↗