Georgia Institute of Technology College of Sciences

Recall that, for a variety $X$ in a projective space $\mathbb{P}^d$, the $X$-rank of a point $p\in \mathbb{P}^d$ is the least number of points of $X$ whose span contains the point $p$. Studies about $X$-ranks include some well-known and important results about various tensor ranks. For example, 

• the rank of tensors is the rank with respect to Segre varieties,
• the rank of symmetric tensors, i.e. Waring rank, is the rank with respect to Veronese embeddings, and
• the rank of anti-symmetric tensors is the rank with respect to Grassmannians in its Plücker embedding.  

In this talk, we focus on ranks with respect to Veronese embeddings of a projective line $\mathbb{P}^1$. i.e. symmetric tensor ranks of binary forms. We will discuss how to associate points in $\mathbb{P}^d$ with binary forms and I will introduce apolarity for binary forms which gives an effective method to study Waring ranks of binary forms. We will discuss various ranks on the Veronese embedding and some results on the ranks.