Details

Autor(en) / Beteiligte

• Schäfke, Reinhard
• Singer, Michael

Titel

Consistent systems of linear differential and difference equations

Ist Teil von

• Journal of the European Mathematical Society : JEMS, 2019-01, Vol.21 (9), p.2751-2792

Ort / Verlag

Zuerich, Switzerland: European Mathematical Society Publishing House

Erscheinungsjahr

2019

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• We consider systems of linear differential and difference equations $$\delta Y(x)/dx = A(x) Y(x), \: \sigma s(Y(x)) = B(x)Y(x)$$ with $\delta = \frac{d}{dx}$, $\sigma$ a shift operator $\sigma(x) = x+a$, $q$-dilation operator $\sigma(x) = qx$ or Mahler operator $\sigma(x) = x^p$ and systems of two linear difference equations $$\sigma_1 Y(x) =A(x)Y(x), \: \sigma_2 Y(x) =B(x)Y(x)$$ with $(\sigma_1,\sigma_2)$ a sufficiently independent pair of shift operators, pair of $q$-dilation operators or pair of Mahler operators. Here $A(x)$ and $B(x)$ are $n\times n$ matrices with rational function entries. Assuming a consistency hypothesis, we show that such systems can be reduced to a system of a very simple form. Using this we characterize functions satisfying two linear scalar differential or difference equations with respect to these operators. We also indicate how these results have consequences both in the theory of automatic sets, leading to a new proof of Cobham's Theorem, and in the Galois theories of linear difference and differential equations, leading to hypertranscendence results.