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3 change the title

# WhatisknownaboutarithmeticDiophantinetheory of homogeneous cubic forms?polynomials

2 couple of typos

# What is knowknown about arithmetic of cubic forms?

Arithmetic of quadratic forms over $\mathbb{Z}$ (or lattices theory) have has received much attention and there are many applications in broad area of mathematics (such as intersection forms on fourfolds). I now wonder whether or not a similar theory for cubic forms can be developed. I recently found a beautiful theorem about binary cubic forms:

Theorem (B. N. Delone and D. K. Faddeev,W.-T. Gan, B. H. Gross, and G. Savin) There is a canonical bijection between isomorphism classes of cubic rings and the set of $GL_{2}(\mathbb{Z})$-equivalence classes of integral binary cubic forms. Under this bijection, the discriminant of a cubic ring is equal to the discriminant of the corresponding binary cubic form.

I don't know how useful this theorem is because I don't know how difficult to classify cubic rings. My question is, are there any classification theory when the number of variable is small? I would appreciate it if anyone could give me a reference for recent development of cubic forms.

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# What is know about arithmetic of cubic forms?

Arithmetic of quadratic forms over $\mathbb{Z}$ (or lattices theory) have received much attention and there are many applications in broad area of mathematics (such as intersection forms on fourfolds). I now wonder whether or not a similar theory for cubic forms can be developed. I recently found a beautiful theorem about binary cubic forms:

Theorem (B. N. Delone and D. K. Faddeev,W.-T. Gan, B. H. Gross, and G. Savin) There is a canonical bijection between isomorphism classes of cubic rings and the set of $GL_{2}(\mathbb{Z})$-equivalence classes of integral binary cubic forms. Under this bijection, the discriminant of a cubic ring is equal to the discriminant of the corresponding binary cubic form.

I don't know how useful this theorem is because I don't know how difficult to classify cubic rings. My question is, are there any classification theory when the number of variable is small? I would appreciate it if anyone could give me a reference for recent development of cubic forms.