Existential Closedness Conjecture
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically in the fields of
model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
and
complex geometry In mathematics, complex geometry is the study of geometry, geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of space (mathematics), spaces su ...
, 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 The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one comp ...
and
Hilbert's Nullstellensatz In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ge ...
which are about solvability of (systems of) polynomial equations in the complex numbers. The conjecture was first proposed by
Boris Zilber Boris Zilber (, born 1949) is a Soviet-British mathematician who works in mathematical logic, specifically model theory. He is a emeritus professor of mathematical logic at the University of Oxford. He obtained his doctorate (Candidate of Scien ...
in his work on the model theory of complex exponentiation.transcendental function In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction ...
involved is the complex exponential function. It was later generalised to exponential functions of semiabelian varieties, and analogous conjectures were proposed for modular functionsexponential function \exp : \mathbb \to \mathbb^: 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 In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture about the transcendence degree of certain field extensions of the rational numbers \mathbb, which would establish the transcendence of a large class ...
. The latter is a long-standing
open problem In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is kno ...
in
transcendental number theory Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways. Transcendenc ...
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_2^+(\mathbb) – 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.Schanuel's conjecture In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture about the transcendence degree of certain field extensions of the rational numbers \mathbb, which would establish the transcendence of a large class ...

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. This is the approach used by Zilber. 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. However, Existential Closedness is a natural conjecture in its own right and makes sense without necessarily assuming Schanuel's conjecture (or any other conjecture). In fact, Schanuel's conjecture is considered out of reach while Existential Closedness seems to be much more tractable as evidenced by recent developments, some of which are discussed below.


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..


See also

*
Schanuel's conjecture In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture about the transcendence degree of certain field extensions of the rational numbers \mathbb, which would establish the transcendence of a large class ...
* Zilber-Pink conjecture *
Boris Zilber Boris Zilber (, born 1949) is a Soviet-British mathematician who works in mathematical logic, specifically model theory. He is a emeritus professor of mathematical logic at the University of Oxford. He obtained his doctorate (Candidate of Scien ...


References

{{Reflist Conjectures Complex geometry Model theory