following the categorical approach to universal algebra through algebraic theories, proposed by f.~w.~lawvere in his phd thesis, this paper aims at introducing a similar setting for general topology. the cornerstone of the new framework is the notion of emph{categorically-algebraic} (emph{catalg}) emph{topological theory}, whose models induce a category of topological structures. we introduce t...

The coamoeba of any complex algebraic plane curve V is its image in the real torus under the argument map. The area counted with multiplicity of the coamoeba of any algebraic curve in (C∗)2 is bounded in terms of the degree of the curve. We show in this Note that up to multiplication by a constant in (C∗)2, the complex algebraic plane curves whose coamoebas are of maximal area (counted with mul...

We introduce and study a transfer map between ideals of the universal enveloping algebras of two members of a reductive dual pair of Lie algebras. Its definition is motivated by the approach to the real Howe duality through the theory of Capelli identities. We prove that this map provides a lower bound on the annihilators of theta lifts of representations with a fixed annihilator ideal. We also...

We prove that the Bergman kernel function associated to a finitely connected domain Ω in the plane is given as a rational combination of only three basic functions of one complex variable: an Alhfors map, its derivative, and one other function whose existence is deduced by means of the field of meromorphic functions on the double of Ω. Because many other functions of conformal mapping and poten...

Zonotopal algebra interweaves algebraic, geometric and combinatorial properties of a given linear map X. Of basic significance in this theory is the fact that the algebraic structures are derived from the geometry (via a nonlinear procedure known as “the least map”), and that the statistics of the algebraic structures (e.g., the Hilbert series of various polynomial ideals) are combinatorial, i....

In this paper, we establish two results concerning algebraic (C,+)-actions on C. First of all, let φ be an algebraic (C,+)-action on C3. By a result of Miyanishi, its ring of invariants is isomorphic to C[t1, t2]. If f1, f2 generate this ring, the quotient map of φ is the map F : C3 → C2, x 7→ (f1(x), f2(x)). By using some topological arguments, we prove that F is always surjective. Secondly we...

Let C be an algebraic curve of genus g ≥ 3. Let us assume that C is not hyperelliptic, so that it is isomorphic to its image by the canonical map φ : C → P. Enriques proved in [1] that φ(C) is the intersection of the quadrics that contain it, except when C is trigonal (that is, it has a g 3) or C is isomorphic to a plane quintic (g = 6). The proof was completed by Babbage [2], and later Petri p...

In this paper, I will prove that assuming Schanuel’s conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any exponential polynomial. This implies a special case of Shapiro’s conjecture: if p(x) and q(x) are two exponential polynomials over the algebraic...

An amoeba is the image of a subvariety an algebraic torus under logarithmic moment map. We consider some qualitative aspects amoebas, establishing results and posing problems for further study. These include determining dimension amoeba, describing as semi-algebraic set, identifying varieties whose amoebas are finite intersection hypersurfaces. show that not full such if its variety nondegenera...