Existential Closedness conjecture
In mathematics, specifically in the fields of model theory and complex geometry, the Existential Closedness conjecture is a statement predicting when systems of equations involving addition, multiplication, and certain transcendental functions have solutions in the complex numbers.
It can be seen as a conjectural generalisation of the Fundamental Theorem of Algebra and Hilbert's Nullstellensatz which are about solvability of (systems of) polynomial equations in the complex numbers.
The conjecture was first proposed by Boris Zilber in his work on the model theory of complex exponentiation.
[1][2] Zilber's conjecture is known as Exponential Closedness or Exponential Algebraic Closedness and covers the case of Existential Closedness when the transcendental function involved is the complex exponential function.
It was later generalised to exponential functions of semiabelian varieties,[3] and analogous conjectures were proposed for modular functions[4] and Shimura varieties.
[5] Informally, given a complex transcendental function
states that systems of equations involving field operations and
unless the existence of a solution would obviously contradict the (hypothetical) algebraic and transcendental properties of
, the algebraic property referred to above is given by the identity
Its transcendental properties are assumed to be captured by Schanuel's conjecture.
The latter is a long-standing open problem in transcendental number theory and implies in particular that
Some systems of equations cannot have solutions because of these properties.
has no solutions, and similarly for any non-zero polynomial
Exponential Closedness states that a system of equations, which is not overdetermined and which cannot be reduced to an overdetermined system by using the above-mentioned algebraic property of
Formally, every free and rotund system of exponential equations has a solution.
Freeness and rotundity are technical conditions capturing the notion of a non-overdetermined system.
In the modular setting the transcendental function under consideration is the
Its algebraic properties are governed by the transformation rules under the action of
rational matrices with positive determinant – on the upper half-plane.
[4] Modular Existential Closedness states that every free and broad system of equations involving field operations and the
Existential Closedness can be seen as a dual statement to Schanuel's conjecture or its analogue in the appropriate setting.
Schanuel implies that certain systems of equations cannot have solutions (or solutions which are independent in some sense, e.g. linearly independent) as the above example of exponential equations demonstrates.
Then Existential Closedness can be interpreted roughly as stating that solutions exists unless their existence would contradict Schanuel's conjecture.
[2] His axiomatisation of pseudo-exponentiation prominently features Schanuel and a strong version of Existential Closedness which is indeed dual to Schanuel.
This strong version predicts existence of generic solutions and follows from the combination of the Existential Closedness, Schanuel, and Zilber-Pink conjectures.
In fact, Schanuel's conjecture is considered out of reach[8] while Existential Closedness seems to be much more tractable as evidenced by recent developments, some of which are discussed below.
The Existential Closedness conjecture is open in full generality both in the exponential and modular settings, but many special cases and weak versions have been proven.
For instance, the conjecture (in both settings) has been proven assuming dominant projection: any system of polynomial equations in the variables
[9][6][10] Another important special case is the solvability of systems of raising to powers type.
[11] Differential/functional analogues of the Existential Closedness conjecture have also been proven.
