The 𝐷-resultant, singularities and the degree of unfaithfulness

Abstract

We introduce the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-resultant of two polynomials in one variable and show how it can be used to decide if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k left-parenthesis f left-parenthesis t right-parenthesis comma g left-parenthesis t right-parenthesis right-parenthesis equals k left-parenthesis t right-parenthesis comma k left-bracket f left-parenthesis t right-parenthesis comma g left-parenthesis t right-parenthesis right-bracket equals k left-bracket t right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">k(f(t),g(t))=k(t),k[f(t),g(t)]=k[t]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and to find the singularities of the curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x equals f left-parenthesis t right-parenthesis comma y equals g left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">x=f(t),y=g(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The second criterion is used to give a very short proof of a special case of the epimorphism theorem of Abhyankar and Moh.

Locations

• Proceedings of the American Mathematical Society - View - PDF