Interpolation determinants of exponential polynomials (Q2707223)

For any pair of integral \(N\)-tuples \(K=(K_1, \dots , K_N)\) and \(t=(t_1, \dots , t_N)\), any pair of \(N\)-tuples of complex numbers \(X=(X_1, \dots , X_N)\) and \(Y=(Y_1, \dots , Y_N)\) denote by \(\Delta _{K,t}(X, Y)\) the determinant NEWLINE\[NEWLINE\Delta _{K,t}(X,Y):=\det\Biggl(\sum^{\min(K_i,t_j)}_{l=0}\frac{1}{l! (K_i-l)!(t_j-l)!} X^{t_j-l}_i Y^{K_i-l}_j e^{X_i Y_j} \Biggr)_{\substack{ 1\leq i\leq N\\ 1\leq j\leq N }}.NEWLINE\]NEWLINE In the paper the author expands determinants of this form as a Taylor series. From this he obtains precise upper bounds for \(|\Delta_{K,t}(X,Y)|\). Further, under some weak conditions, the non-vanishing of \(\Delta_{K,t}(X,Y)\) is proved. Combining the results a new simple proof of the Gel'fond-Schneider-Theorem in the real case is obtained.