Mathematicians extend 150-year-old curve theory to new computational landscapes
Researchers have generalized Bring's curve—a foundational object in algebraic geometry—to work across any mathematical field, removing longstanding computational constraints. The advance opens pathways for cryptography, coding theory, and computational algebra applications that previously faced theoretical bottlenecks.
Originaltitel: A generalization of Bring’s curve in any characteristic
Abstract A natural generalization of Bring’s curve, valid over any field 𝕂 of characteristic zero or characteristic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>7</m:mn> </m:mrow> </m:math> p\geq 7 , is the algebraic variety 𝑉 of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>PG</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{PG}(m-1,\mathbb{K}) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mn>5</m:mn> </m:mrow> </m:math> m\geq 5 , which is the complete intersection of the projective algebraic hypersurfaces of the homogeneous equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msubsup> <m:mi>x</m:mi> <m:mn>1</m:mn> <m:mi>k</m:mi> </m:msubsup> <m:mo>+</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo>+</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mi>m</m:mi> <m:mi>k</m:mi> </m:msubsup> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> x_{1}^{k}+\cdots+x_{m}^{k}=0 with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>k</m:mi> <m:mo>≤</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> 1\leq k\leq m-2 . In positive characteristic, we also assume <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>≤</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> m\leq p-1 . Up to a change of coordinates in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>PG</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{PG}(m-1,\mathbb{K}) , we show that 𝑉 is a projective, absolutely irreducible, non-singular curve of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>PG</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{PG}(m-2,\mathbb{K}) . We show that if the automorphism group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Aut</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>V</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{Aut}(V) is tame (in particular in characteristic zero), then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">