@article{Li:2006aa,
    Abstract = {Let {\$}{\$}F={\{}{$\backslash$}cal C{\}}(x{\\_}1,x{\\_}2, {$\backslash$}cdots, x{\\_}{$\backslash$}ell, x{\\_}{\{}{$\backslash$}ell+1{\}}, {$\backslash$}cdots, x{\\_}m){\$}{\$}, where {\$}{\$}x{\\_}1,x{\\_}2, {$\backslash$}cdots, x{\\_}{$\backslash$}ell{\$}{\$}are differential variables, and {\$}{\$}x{\\_}{\{}{$\backslash$}ell+1{\}}, {$\backslash$}cdots, x{\\_}m{\$}{\$}are shift variables. We show that a hyperexponential function, which is algebraic over {\$}{\$}F{\$}{\$}, is of form{\$}{\$}g(x{\\_}1,x{\\_}2, {$\backslash$}cdots, x{\\_}m) q(x{\\_}1,x{\\_}2, {$\backslash$}cdots, x{\\_}{$\backslash$}ell)\^{}{$\backslash$}frac{\{}1{\}}{\{}t{\}}{$\backslash$}omega{\\_}{\{}{$\backslash$}ell+1{\}}\^{}{\{}x{\\_}{\{}{$\backslash$}ell+1{\}}{\}} {$\backslash$}cdots {$\backslash$}omega{\\_}m\^{}{\{}x{\\_}m{\}},{\$}{\$}where {\$}{\$}g {$\backslash$}in F, q {$\backslash$}in {\{}{$\backslash$}cal C{\}}(x{\\_}1,x{\\_}2, {$\backslash$}cdots, x{\\_}{$\backslash$}ell), t {$\backslash$}in {\{}{$\backslash$}cal Z{\}}\^{}+{\$}{\$}and {\$}{\$}{$\backslash$}omega{\\_}{\{}{$\backslash$}ell+1{\}}, {$\backslash$}cdots, {$\backslash$}omega{\\_}m{\$}{\$}are roots of unity. Furthermore, we present an algorithm for determining whether a hyperexponential function is algebraic over {\$}{\$}F{\$}{\$}.},
    Author = {Li, Ziming and Zheng, Dabin},
    Date = {2006/09/01},
    File = {Determining Whether a Multivariate Hyperexponential Function is Algebraic - s11424-006-0352-5.pdf},
    ISBN = {1559-7067},
    Journal = {Journal of Systems Science and Complexity},
    Number = {3},
    Pages = {352--364},
    Title = {Determining Whether a Multivariate Hyperexponential Function is Algebraic},
    URL = {https://doi.org/10.1007/s11424-006-0352-5},
    Volume = {19},
    Year = {2006},
    bdsk-url-1 = {https://doi.org/10.1007/s11424-006-0352-5},
    date-added = {2022-12-16 22:30:46 +0100},
    date-modified = {2022-12-16 22:30:46 +0100},
    id = {Li2006},
    doi = {10.1007/s11424-006-0352-5}
}

@article{Li:2006aa, Abstract = {Let {\$}{\$}F={{}{$\backslash$}cal C{}}(x{\}1,x{\}2, {$\backslash$}cdots, x{\}{$\backslash$}ell, x{\}{{}{$\backslash$}ell+1{}}, {$\backslash$}cdots, x{\}m){\$}{\$}, where {\$}{\$}x{\}1,x{\}2, {$\backslash$}cdots, x{\}{$\backslash$}ell{\$}{\$}are differential variables, and {\$}{\$}x{\}{{}{$\backslash$}ell+1{}}, {$\backslash$}cdots, x{\}m{\$}{\$}are shift variables. We show that a hyperexponential function, which is algebraic over {\$}{\$}F{\$}{\$}, is of form{\$}{\$}g(x{\}1,x{\}2, {$\backslash$}cdots, x{\}m) q(x{\}1,x{\}2, {$\backslash$}cdots, x{\}{$\backslash$}ell)\^{}{$\backslash$}frac{{}1{}}{{}t{}}{$\backslash$}omega{\}{{}{$\backslash$}ell+1{}}\^{}{{}x{\}{{}{$\backslash$}ell+1{}}{}} {$\backslash$}cdots {$\backslash$}omega{\}m\^{}{{}x{\}m{}},{\$}{\$}where {\$}{\$}g {$\backslash$}in F, q {$\backslash$}in {{}{$\backslash$}cal C{}}(x{\}1,x{\}2, {$\backslash$}cdots, x{\}{$\backslash$}ell), t {$\backslash$}in {{}{$\backslash$}cal Z{}}\^{}+{\$}{\$}and {\$}{\$}{$\backslash$}omega{\}{{}{$\backslash$}ell+1{}}, {$\backslash$}cdots, {$\backslash$}omega{\_}m{\$}{\$}are roots of unity. Furthermore, we present an algorithm for determining whether a hyperexponential function is algebraic over {\$}{\$}F{\$}{\$}.}, Author = {Li, Ziming and Zheng, Dabin}, Date = {2006/09/01}, File = {Determining Whether a Multivariate Hyperexponential Function is Algebraic - s11424-006-0352-5.pdf}, ISBN = {1559-7067}, Journal = {Journal of Systems Science and Complexity}, Number = {3}, Pages = {352--364}, Title = {Determining Whether a Multivariate Hyperexponential Function is Algebraic}, URL = {https://doi.org/10.1007/s11424-006-0352-5}, Volume = {19}, Year = {2006}, bdsk-url-1 = {https://doi.org/10.1007/s11424-006-0352-5}, date-added = {2022-12-16 22:30:46 +0100}, date-modified = {2022-12-16 22:30:46 +0100}, id = {Li2006}, doi = {10.1007/s11424-006-0352-5} }