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Adam Parusinski

Publications and source records attributed to Adam Parusinski.

At least 37 records · Page 2Linked to original sources
arXiv2015

Lifting differentiable curves from orbit spaces

Let $ρ: G \rightarrow \operatorname{O}(V)$ be a real finite dimensional orthogonal representation of a compact Lie group, let $σ= (σ_1,\ldots,σ_n) : V \to \mathbb R^n$, where $σ_1,\ldots,σ_n$ form a minimal system of homogeneous generators of the $G$-invariant polynomials on $V$, and set $d = \max_i \operatorname{deg} σ_i$. We prove that for each $C^{d-1,1}$-curve $c$ in $σ(V) \subseteq \mathbb R^n$ there exits a locally Lipschitz lift over $σ$, i.e., a locally Lipschitz curve $\overline c$ in $V$ so that $c = σ\circ \overline c$, and we obtain explicit bounds for the Lipschitz constant of $\overline c$ in terms of $c$. Moreover, we show that each $C^d$-curve in $σ(V)$ admits a $C^1$-lift. For finite groups $G$ we deduce a multivariable version and some further results.

math.DG↗
arXiv2014

Regular Covers for Open Relatively Compact Subanalytic Sets

Let $U$ be an open relatively compact subanalytic subset of a real analytic manifold. We show that there exists a finite linear covering (in the sense of Guillermou and Schapira) of $U$ by subanalytic open subsets of $U$ homeomorphic to a unit ball. We also show that the algebra of open relatively compact subanalytic subsets of a real analytic manifold is generated by subsets subanalytically and bi-lipschitz homeomorphic to a unit ball.

math.AG↗
arXiv2014

Local topological algebraicity of analytic function germs

T. Mostowski showed that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that every (real or complex) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety.

math.AG↗
arXiv2012

The weight filtration for real algebraic varieties II: Classical homology

We associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on classical (compactly supported) homology with Z/2 coefficients an analog of the weight filtration for complex algebraic varieties. This complements our previous definition of the weight filtration of Borel-Moore homology.

math.AG↗
arXiv2012

The Abhyankar-Jung Theorem

We show that every quasi-ordinary Weierstrass polynomial $P(Z) = Z^d+a_1 (X) Z^{d-1}+...+a_d(X) \in \K[[X]][Z] $, $X=(X_1,..., X_n)$, over an algebraically closed field of characterisic zero $\K$, and satisfying $a_1=0$, is $ν$-quasi-ordinary. That means that if the discriminant $Δ_P \in \K[[X]]$ is equal to a monomial times a unit then the ideal $(a_i^{d!/i}(X))_{i=2,...,d}$ is principal and generated by a monomial. We use this result to give a constructive proof of the Abhyankar-Jung Theorem that works for any Henselian local subring of $\K[[X]]$ and the function germs of quasi-analytic families.

math.AC↗
arXiv2009

The weight filtration for real algebraic varieties

Using the work of Guillen and Navarro Aznar we associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on Borel-Moore homology with Z/2 coefficients an analog of the weight filtration for complex algebraic varieties.

math.AG↗
arXiv2009

On the non-analyticity locus of an arc-analytic function

In this paper we show that the non-analyticity locus of an arc-analytic function is arc-symmetric. Recall that a function is called arc-analytic if it is real analytic on each real analytic arc. By a result of Bierstone and Milman a big class of arc-analytic function, namely those that satisfy a polynomial equation with real analytic coefficients, can be made analytic by a sequence of global blowings-up with smooth centers. We show that these centers can be chosen, at each stage of the resolution, inside the non-analyticity locus.

math.AG↗
arXiv2008

A criterion for topological equivalence of two variable complex analytic function germs

We show that two analytic function germs $(\C^2,0) \to (\C,0)$ are topologically right equivalent if and only if there is a one-to-one correspondence between the irreducible components of their zero sets that preserves the multiplicites of these components, their Puiseux pairs, and the intersection numbers of any pairs of distinct components.

math.AG↗
arXiv2008

Blow-analytic equivalence of two variable real analytic function germs

Blow-analytic equivalence is a notion for real analytic function germs, introduced by Tzee-Char Kuo in order to develop real analytic equisingularity theory. In this paper we give complete characterisations of blow-analytic equivalence in the two dimensional case: in terms of the real tree model for the arrangement of real parts of Newton-Puiseux roots and their Puiseux pairs, and in terms of minimal resolutions. These characterisations show that in the two dimensional case the blow-analytic equivalence is a natural analogue of topological equivalence of complex analytic function germs. Moreover, we show that in the two-dimensional case the blow-analytic equivalence can be made cascade, and hence satisfies several geometric properties. It preserves, for instance, the contact orders of real analytic arcs. In the general $n$-dimensional case, we show that a singular real modification satisfies the arc-lifting property.

math.AG↗
arXiv2008

Equivalence relations for two variable real analytic function germs

For two variable real analytic function germs we compare the blow-analytic equivalence in the sense of Kuo to the other natural equivalence relations. Our main theorem states that $C^1$ equivalent germs are blow-analytically equivalent. This gives a negative answer to a conjecture of Kuo. In the proof we show that the Puiseux pairs of real Newton-Puiseux roots are preserved by the $C^1$ equivalence of function germs. The proof is achieved, being based on a combinatorial characterisation of blow-analytic equivalence in terms of the real tree model. We also give several examples of bi-Lipschitz equivalent germs that are not blow-analytically equivalent.

math.AG↗
arXiv2002

Motivic-type Invariants of Blow-analytic Equivalence

To a given analytic function germ $f:(\mathbb{R}^d,0) \to (\mathbb{R},0)$, we associate zeta functions $Z_{f,+}$, $Z_{f,-} \in \mathbb{Z} [[T]]$, defined analogously to the motivic zeta functions of Denef and Loeser. We show that our zeta functions are rational and that they are invariants of the blow-analytic equivalence in the sense of Kuo. Then we use them together with the Fukui invariant to classify the blow-analytic equivalence classes of Brieskorn polynomials of two variables. Except special series of singularities our method classifies as well the blow-analytic equivalence classes of Brieskorn polynomials of three variables.

math.AG↗
arXiv2002

Newton-Puiseux Roots of Jacobian Determinants

Let $f(x,y), g(x,y)$ denote either a pair of holomorphic function germs, or a pair of monic polynomials in $x$ whose coefficients are Laurent series in $y$. A relative polar arc is a Newton-Puiseux root, $x=γ(y)$, of the Jacobian $J=f_yg_x-f_xg_y$. We define the tree-model, $T(f,g)$, for the pair, using the contact orders of the Newton-Puiseux roots of $f$ and $g$. We then describe how the $γ$'s climb, and where they leave, the tree. We shall also show by two examples that the way the $γ$'s leave the tree is not an invariant of the tree; this phenomenon is in sharp contrast to that in the one function case where the tree completely determines how the polar roots split away. Our result yield a factorisation of the Jacobian determinant.

math.AG↗
arXiv2002

Topology of Injective Endomorphisms of Real Algebraic Sets

Using only basic topological properties of real algebraic sets and regular morphisms we show that any injective regular self-mapping of a real algebraic set is surjective. Then we show that injective morphisms between germs of real algebraic sets define a partial order on the equivalence classes of these germs divided by continuous semi-algebraic homeomorphisms. We use this observation to deduce that any injective regular self-mapping of a real algebraic set is a homeomorphism. We show also a similar local property. All our results can be extended to arc-symmetric semi-algebraic sets and injective continuous arc-symmetric morphisms, and some results to Euler semi-algebraic sets and injective continuous semi-algebraic morphisms.

math.AG↗
arXiv2002

Virtual Betti numbers of real algebraic varieties

The weak factorization theorem for birational maps is used to prove that for all nonnegative i the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a "virtual Betti number" beta_i defined for all real algebraic varieties, such that if Y is a closed subvariety of X, then beta_i(X) = beta_i(X\Y) + beta_i(Y).

math.AG↗
arXiv2002

Algebraically constructible functions: real algebra and topology

Algebraically constructible functions connect real algebra with the topology of algebraic sets. In this survey we present some history, definitions, properties, and algebraic characterizations of algebraically constructible functions, and a description of local obstructions for a topological space to be homeomorphic to a real algebraic set.

math.AG↗
arXiv2000

Proof of the gradient conjecture of R. Thom

Let x(t) be a trajectory of the gradient of a real analytic function and suppose that x_0 is a limit point of x(t). We prove the gradient conjecture of R. Thom which states that the secants of x(t) at x_0 have a limit. Actually we show a stronger statement: the radial projection of x(t) from x_0 onto the unit sphere has finite length.

math.AG↗