Gauss Composition? and a naive composition law

[Mathguy15]

What exactly is gauss composition? I've heard of Manjul Bhargava's work, which apparently generalized gauss composition, but what is gauss composition? I would like to add that I've been thinking about quadratics polynomials with rational coefficients, and I discovered this composition law that turns the set of quadratic polynomials into an abelian group. Let f(x)=ax^2+bx+c and g(x)=zx^2+dx+r be two quadratic polynomials with rational coefficients. Denote the set of quadratic polynomials with rational coefficients by T{x}. Then the composition law %:T{x} X T{x}--->T{x} defined by f(x)%g(x)=azx^2+bdx+cr turns T{x} into an abelian group. This has probably already been figured out before, but an interesting note!
mathguy

EDIT:(simple explanations please, thank you.)

 

[Stephen Tashi]

Getting a good answer to a question is somewhat a matter of luck. There might be an expert on Gauss composition on the forum who is chomping at the bit to answer such a very general question. If no such expert turns up, I suggest you ask a more specific question. This PDF looks interesting: http://www.google.com/url?sa=t&rct=...sg=AFQjCNFCUMwAwetrjbw_3lkt373P3ppmJQ&cad=rja

It tells what Gauss thought that Gauss composition was. If you have a specific question about something in it, you might lure me or some other non-Gauss-composition student into reading it and trying to answer. (I haven't read it yet.)

According to that PDF, Gauss composition is a ternary operation, not a binary operation. As to the Abelian group idea, how are you going to define inverses?

 

[lostcauses10x]

"simple explanations please"

No simple explanation that I can find.

A few papers that might explain it to you can be found such as
The shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae

A copy of Disquisitiones Arithmeticae converted to English can also be had, though a bit pricy.

 

[Mathguy15]

Getting a good answer to a question is somewhat a matter of luck. There might be an expert on Gauss composition on the forum who is chomping at the bit to answer such a very general question. If no such expert turns up, I suggest you ask a more specific question. This PDF looks interesting: http://www.google.com/url?sa=t&rct=...sg=AFQjCNFCUMwAwetrjbw_3lkt373P3ppmJQ&cad=rja

It tells what Gauss thought that Gauss composition was. If you have a specific question about something in it, you might lure me or some other non-Gauss-composition student into reading it and trying to answer. (I haven't read it yet.)

According to that PDF, Gauss composition is a ternary operation, not a binary operation. As to the Abelian group idea, how are you going to define inverses?

ok, so its a ternary operation rather than a binary. I will look into that pdf you have. With regards to the abelian group idea, let f(x)=ax^2+bx+c be a quadratic polynomial with rational coefficients. Let g(x)=(1/a)x^2+(1/b)x+1/c. Then, f(x)%g(x)=a(1/a)x^2+b(1/b)x+c(1/c)=x^2+x+1. x^2+x+1 is the identity, because if f(x)=ax^2+bx+c and t(x)=x^2+x+1, then f(x)%t(x)=a(1)x^2+b(1)x+c(1)=ax^2+bx+c=f(x), and t(x)%f(x)=1ax^2+1bx+1c=ax^2+bx+c=f(x).

EDIT: I see now. 0 can't be one of the coefficients. So if f(x)=ax^2+bx+c AND if neither b nor c equals 0, then the set along with the naive composition forms an abelian group.

EDIT(again): In http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_13.pdf, Bhargava says that Gauss laid down a remarkable law of composition on integral binary quadratic forms. Did he find several?

 

[blue_raver22]

Gauss composition is a mathematical concept that was first introduced by mathematician Carl Friedrich Gauss. It involves combining two polynomials to create a new polynomial. This process is also known as polynomial composition.

The naive composition law that you have discovered is a specific type of Gauss composition for quadratic polynomials with rational coefficients. It involves multiplying the leading coefficients and adding the constant terms of the two polynomials to create a new polynomial.

This composition law turns the set of quadratic polynomials with rational coefficients into an abelian group, which means that it follows the properties of an abelian group such as closure, associativity, identity element, and inverse elements.

While this concept may have been previously discovered, your observation is still interesting and could potentially have applications in other areas of mathematics.