I am currently writing my dissertation on elementary functions and elementary numbers (in the sense of Ritt and Liouville etc.). The definition of an elementary number I am using is as follows;
We say that an elementary number is a number of the form $p(a_{1},...,a_{n})$ where $p$ is an exponential polynomial and $(a_{1},...,a_{n})$ is a non-singular solution of a system $A=0$ where $A$ is a system of $n$ exponential polynomial equations in $n$ unknowns.
Note here we are defining an exponential polynomial in variables $x_{1},...,x_{n}$ to be a polynomial with rational coefficients in $x_{1},...,x_{n},e^{x_{1}},...,e^{x_{n}}$ that is to say an element of $\mathbb{Q}[x_{1},...,x_{n},e^{x_{1}},...,e^{x_{n}}]$, and if we now consider a system of n exponential polynomials $A$, in variables $x_{1},...,x_{n}$, we say that a solution in $\mathbb{C}^{n}$ of such a system $A=0$ is non-singular if at such a point the Jacobian $J(A)$ is non-singular, that is to say the Jacobian has a non-zero determinant and as such is invertible.
Can someone explain to me if this definition is equivalent to the one in the following article: http://delivery.acm.org/10.1145/230000/220360/p104-richardson.pdf?ip=139.222.81.70&id=220360&acc=ACTIVE%20SERVICE&key=BF07A2EE685417C5%2E548157F5A5FEB351%2E4D4702B0C3E38B35%2E4D4702B0C3E38B35&CFID=500761947&CFTOKEN=79878506&__acm__=1428931019_48f37e627dccb1913c7243c5adf1ae90 if so how?
And how can I use my definition to prove that these elementary numbers form an algebrically closed field and are closed under application of elementary functions, where elementary functions are defined as essentially all functions formed from algebraic functions, exponentiation and taking logs all in base e over the complex numbers?