Regular extensions and algebraic relations between values of Mahler functions in positive characteristic

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper K"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a function field of characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We have recently established the analogue of a theorem of Ku. Nishioka for linear Mahler systems defined over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper K left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {K}(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This paper is dedicated to proving the following refinement of this theorem. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f 1 left-parenthesis z right-parenthesis comma ellipsis comma f Subscript n Baseline left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f_{1}(z),\ldots , f_{n}(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Mahler functions such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove double-struck upper K With bar left-parenthesis z right-parenthesis left-parenthesis f 1 left-parenthesis z right-parenthesis comma ellipsis comma f Subscript n Baseline left-parenthesis z right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\overline {\mathbb {K}}(z)\left (f_{1}(z),\ldots , f_{n}(z)\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a regular extension over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove double-struck upper K With bar left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\overline {\mathbb {K}}(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Then, every homogeneous algebraic relation over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper K overbar"> <mml:semantics> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex">\overline {\mathbb {K}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> between their values at a regular algebraic point arises as the specialization of a homogeneous algebraic relation over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove double-struck upper K With bar left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\overline {\mathbb {K}}(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> between these functions themselves. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper K"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is replaced by a number field, this result is due to B. Adamczewski and C. Faverjon as a consequence of a theorem of P. Philippon. The main difference is that in characteristic zero, every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Mahler extension is regular, whereas in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, non-regular <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Mahler extensions do exist. Furthermore, we prove that the regularity of the field extension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove double-struck upper K With bar left-parenthesis z right-parenthesis left-parenthesis f 1 left-parenthesis z right-parenthesis comma ellipsis comma f Subscript n Baseline left-parenthesis z right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\overline {\mathbb {K}}(z)\left (f_{1}(z),\ldots , f_{n}(z)\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is also necessary for our refinement to hold. Besides, we show that when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p does-not-divide d"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∤<!-- ∤ --></mml:mo> <mml:mi>d</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p\nmid d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Mahler extensions over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove double-struck upper K With bar left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\overline {\mathbb {K}}(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are always regular. Finally, we describe some consequences of our main result concerning the transcendence of values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Mahler functions at algebraic points.