Understanding Quartic Curves in Classical Algebraic Geometry

Algebraic geometry is a captivating branch of mathematics that studies solutions to polynomial equations. One of the fundamental objects of study in this field is algebraic curves, which are defined by polynomial equations in two variables. Among these curves, the quartic curve holds a special place. Let’s delve into its definition, characteristics, and significance. <h1>What is a Quartic Curve?</h1> A quartic curve in classical algebraic geometry is an algebraic curve defined by a polynomial equation of degree four. In the plane, this curve can be represented by an equation of the form: F(x,y)=a0+a1x+a2y+a3x^2+a4xy+a5y^2+a6x^3+a7x^2y+a8xy^2+a9y^3+a10x^4+a11x^3y+a12x^2y^2+a13xy^3+a14y^4=0 where at least one of the coefficients a10,a11,a12,a13,a14 is non-zero. This ensures that the curve is genuinely of degree four. <a href="https://medium.com/@andrew_johnson_4/understanding-quartic-curves-in-classical-algebraic-geometry-c2b889579799">Website</a>