• Corpus ID: 250089071

On Some Systems of Equations in Abelian Varieties

@inproceedings{Gallinaro2022OnSS,
  title={On Some Systems of Equations in Abelian Varieties},
  author={Francesco Paolo Gallinaro},
  year={2022}
}
We solve a case of the Abelian Exponential-Algebraic Closedness Conjecture , a conjecture due to Bays and Kirby, building on work of Zilber, which predicts sufficient conditions for systems of equations involving algebraic operations and the exponential map of an abelian variety to be solvable in the complex numbers. More precisely, we show that the conjecture holds for subvarieties of the tangent bundle of an abelian variety A which split as the product of a linear subspace of the Lie algebra of… 

Figures from this paper

References

SHOWING 1-10 OF 35 REFERENCES

A geometric approach to some systems of exponential equations

Zilber’s Exponential Algebraic Closedness conjecture (also known as Zilber’s Nullstellensatz) gives conditions under which a complex algebraic variety should intersect the graph of the exponential

Exponential Sums Equations and Tropical Geometry

We show a case of Zilber’s Exponential-Algebraic Closedness Conjecture, establishing that the conjecture holds for varieties which split as the product of a linear subspace of the additive group C n

The theory of exponential sums

We consider the theory of algebraically closed fields of characteristic zero with multivalued operations $x\mapsto x^r$ (raising to powers). It is in fact the theory of equations in exponential sums.

Solving Systems of Equations of Raising-to-Powers Type

We address special cases of the analogues of the exponential algebraic closedness conjecture relative to the exponential maps of semiabelian varieties and to the modular j function. In particular, we

Pseudo-exponential maps, variants, and quasiminimality

We give a construction of quasiminimal fields equipped with pseudo-analytic maps, generalising Zilber's pseudo-exponential function. In particular we construct pseudo-exponential maps of simple

Local Properties of Analytic Varieties

Algebraic and analytic varieties have become increasingly important in recent years, both in the complex and the real case. Their local structure has been intensively investigated, by algebraic and

Polynomial–exponential equations and Zilber's conjecture

Assuming Schanuel's conjecture, we prove that any polynomial–exponential equation in one variable must have a solution that is transcendental over a given finitely generated field. With the help of

Algebraic flows on abelian varieties

  • E. UllmoA. Yafaev
  • Mathematics
    Journal für die reine und angewandte Mathematik (Crelles Journal)
  • 2018
Let A be an abelian variety. The abelian Ax–Lindemann theorem shows that the Zariski closure of an algebraic flow in A is a translate of an abelian subvariety of A. The paper discusses some

The theory of the exponential differential equations of semiabelian varieties

The complete first-order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arise from an amalgamation-with-predimension

Algebraic flows on commutative complex Lie groups

We recover results by Ullmo-Yafaev and Peterzil-Starchenko on the closure of the image of an algebraic variety in a compact complex torus. Our approach uses directed closed currents and allows us to