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. 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, and analogous conjectures were proposed for modular functions and Shimura varieties.

Statement

Informally, given a complex transcendental function $f$, the Existential Closedness conjecture for $f$ states that systems of equations involving field operations and $f$ always have solutions in $\mathbb {C}$ unless the existence of a solution would obviously contradict the (hypothetical) algebraic and transcendental properties of $f$. Two precise cases are considered below.

Exponential Closedness

In the case of the exponential function $\exp :\mathbb {C} \to \mathbb {C} ^{\times }:z\mapsto e^{z}$, the algebraic property referred to above is given by the identity $\exp(z_{1}+z_{2})=\exp(z_{1})\cdot \exp(z_{2})$. 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 $e$ and $\pi$ are algebraically independent over the rationals.

Some systems of equations cannot have solutions because of these properties. For instance, the system $z_{2}=2z_{1}+1,\exp(z_{2})=(\exp(z_{1}))^{2}$ has no solutions, and similarly for any non-zero polynomial $p(X,Y)$ with rational coefficients the system $\exp(z)=-1,p(z,\exp(1))=0$ has no solution if we assume $e$ and $\pi$ are algebraically independent. The latter is an example of an overdetermined system, where we have more equations than variables. 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 $\exp$, always has solutions in the complex numbers. 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.

Modular Existential Closedness

In the modular setting the transcendental function under consideration is the $j$-function. Its algebraic properties are governed by the transformation rules under the action of $\mathrm {GL} _{2}^{+}(\mathbb {Q} )$ – the group of $2\times 2$ rational matrices with positive determinant – on the upper half-plane. The transcendental properties of $j$ are captured by the Modular Schanuel Conjecture.

Modular Existential Closedness states that every free and broad system of equations involving field operations and the $j$-function has a complex solution, where freeness and broadness play the role of freeness and rotundity mentioned above.

Partial results and special cases

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 $z_{1},...,z_{n}$ and $\exp(z_{1}),...,\exp(z_{n})$ (or $j(z_{1}),...,j(z_{n})$), which does not imply any algebraic relation between $z_{1},...,z_{n}$, has complex solutions. Another important special case is the solvability of systems of raising to powers type. Differential/functional analogues of the Existential Closedness conjecture have also been proven.