Is a bilinear map $\cdot: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$ with no zero divisors smooth?

Question

No zero divisors as in if $xy = 0$ then $x = 0$ or $y = 0$.

Answer 1

Any bi-linear map of finite dimensional vector spaces is smooth. It is easy to see e.g. by the universal property of tensor product (any bi-linear map lifts to a linear map from the tensor product which is smooth), or just right from the definition (derivatives with respect to every coordinate exist). they actually are given by homogeneous polynomials of degree 2.