# Solving Systems of Equations of Raising-to-Powers Type

@inproceedings{Gallinaro2021SolvingSO, title={Solving Systems of Equations of Raising-to-Powers Type}, author={Francesco Paolo Gallinaro}, year={2021} }

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 show that the graph of the exponential of an abelian variety intersects products of free rotund varieties in which the subvariety of the domain is a sufficiently generic linear subspace, and that the graph of j intersects products of free broad varieties in which the subvariety of the domain is a M…

## 3 Citations

### On Some Systems of Equations in Abelian Varieties

- Mathematics
- 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…

### Algebraic Varieties and Automorphic Functions

- Mathematics
- 2021

Let (G,X) be a Shimura datum, let Ω be a connected component of X, let Γ be a congruence subgroup of G(Q), and consider the quotient map q : Ω → S := Γ\Ω. Consider the Harish-Chandra embedding Ω ⊂ C…

### Solutions of equations involving the modular $j$ function

- MathematicsTransactions of the American Mathematical Society
- 2019

Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular $j$ function. We show general cases in which these systems have solutions,…

## References

SHOWING 1-10 OF 36 REFERENCES

### The theory of the exponential differential equations of semiabelian varieties

- Mathematics
- 2009

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…

### Pseudo-exponentiation on algebraically closed fields of characteristic zero

- MathematicsAnn. Pure Appl. Log.
- 2005

### Algebraic flows on abelian varieties

- MathematicsJournal 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…

### Solutions of equations involving the modular $j$ function

- MathematicsTransactions of the American Mathematical Society
- 2019

Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular $j$ function. We show general cases in which these systems have solutions,…

### The theory of exponential sums

- Mathematics
- 2015

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.…

### Polynomial–exponential equations and Zilber's conjecture

- Mathematics
- 2016

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…

### The theory of exponential differential equations

- Mathematics
- 2006

This thesis is a model-theoretic study of exponential differential equations in the context of differential algebra. I define the theory of a set of differential equations and give an axiomatization…

### Generic solutions of equations with iterated exponentials

- Mathematics
- 2016

We study solutions of exponential polynomials over the complex field. Assuming Schanuel's conjecture we prove that certain polynomials have generic solutions in the complex field.

### Pseudo-exponential maps, variants, and quasiminimality

- MathematicsAlgebra & Number Theory
- 2018

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…

### Differential existential closedness for the 𝑗-function

- MathematicsProceedings of the American Mathematical Society
- 2021

We prove the Existential Closedness conjecture for the differential equation of the
j
j
-function and its derivatives. It states that in a differentially closed field certain equations…