Minimal polynomial decomposition of plane curve branch truncation using a semigroup based algorithm

Abstract Let Y be an irreducible plane curve germ with branch $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> and s characteristic exponents. We introduce a class of truncation sequences of $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> having finite support. For a given $$({\widetilde{\zeta }}_i)_{i=1,\ldots ,s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mover> <mml:mi>ζ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> </mml:math> from this class, we explicitly compute the convex hull of the minimal polynomial $$f_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> for each germ of plane curve $$Y_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> , with branch $${\widetilde{\zeta }}_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>ζ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mi>i</mml:mi> </mml:msub> </mml:math> . We investigate the relationships between the semigroup of the $$Y_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> ’s, as well as the induced canonical valuations. Additionally, we provide methods for selecting truncation sequences that yield topologically equivalent approximations $$Y_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> of Y . The sequence $$({\widetilde{\zeta }}_i)_{i=1,\ldots ,s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mover> <mml:mi>ζ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> </mml:math> of $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> provides a unique decomposition of each polynomial $$f_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> . Given that the minimal polynomial $$f_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> can be written as a power of $$f_{i-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> plus a tail $$\delta _i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>δ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> , our first decomposition theorem studies properties of the tail. The second decomposition theorem characterizes the decomposition of $$\delta _i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>δ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> and enables its explicit computation. To conclude, a pseudocode algorithm is presented along with an example.