Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation

We propose a new method to compute the unipotent radical $R_u(H)$ of the differential Galois group $H$ associated to a parameterized second-order homogeneous linear differential equation of the form $$\tfrac{\partial^2}{\partial x^2}Y-qY=0,$$ where $q \in F(x)$ is a rational function in $x$ with coefficients in a $\Pi$-field $F$ of characteristic zero, and $\Pi$ is a commuting set of parametric derivations. The procedure developed by Dreyfus reduces the computation of $R_u(H)$ to solving a creative telescoping problem, whose effective solution requires the assumption that the maximal reductive quotient $H / R_u(H)$ is a $\Pi$-constant linear differential algebraic group. When this condition is not satisfied, we compute a new set of parametric derivations $\Pi'$ such that the associated differential Galois group $H'$ has the property that $H'/ R_u(H')$ is $\Pi'$-constant, and such that $R_u(H)$ is defined by the same differential equations as $R_u(H')$. Thus the computation of $R_u(H)$ is reduced to the effective computation of $R_u(H')$. We expect that an elaboration of this method will be successful in extending the applicability of some recent algorithms developed by Minchenko, Ovchinnikov, and Singer to compute unipotent radicals for higher order equations.Comment: 12 page

Computing the differential Galois group of a parameterized second-order linear differential equation

We develop algorithms to compute the differential Galois group $G$ associated to a parameterized second-order homogeneous linear differential equation of the form $$\tfrac{\partial^2}{\partial x^2} Y + r_1 \tfrac{\partial}{\partial x} Y + r_0 Y = 0,$$ where the coefficients $r_1, r_0 \in F(x)$ are rational functions in $x$ with coefficients in a partial differential field $F$ of characteristic zero. Our work relies on the procedure developed by Dreyfus to compute $G$ under the assumption that $r_1 = 0$. We show how to complete this procedure to cover the cases where $r_1 \neq 0$, by reinterpreting a classical change of variables procedure in Galois-theoretic terms.Comment: 14 page

The Membership Problem for Ideals in Z[X]

There exists a feasible procedure to decide whether or not an arbitrary polynomial belongs to a given ideal in Z[x] if the ideal\u27s minimal basis is known. However, when this is not the case there is no feasible procedure to decide whether or not an arbitrary polynomial belongs to a given ideal. There already exists an effective procedure to find an ideal\u27s minimal basis, but it depends on solving the membership problem for the ideal (i.e. the problem of deciding whether an arbitrary polynomial belongs to the ideal). Therefore, we develop a modification of the existing algorithm to find an ideal\u27s minimal basis so that there is no need to solve the membership problem to carry it out, and then we use this minimal basis to solve the membership problem for this ideal

Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions

We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental

Twisted Mahler discrete residues

Recently we constructed Mahler discrete residues for rational functions and showed they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $g(x^p)-g(x)$ for some rational function $g(x)$ and an integer $p > 1$. Here we develop a notion of $\lambda$-twisted Mahler discrete residues for $\lambda\in\mathbb{Z}$, and show that they similarly comprise a complete obstruction to the twisted Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $p^\lambda g(x^p)-g(x)$ for some rational function $g(x)$ and an integer $p>1$. We provide some initial applications of twisted Mahler discrete residues to differential creative telescoping problems for Mahler functions and to the differential Galois theory of linear Mahler equations