# The monic rank

## TL;DR

This paper introduces the monic rank, a new measure related to classical rank concepts, and provides algorithms and results that connect it to longstanding conjectures and specific cases in algebraic geometry and invariant theory.

## Contribution

It defines the monic rank, proves its finiteness, develops an algorithmic approach to compute maximal monic rank, and verifies conjectures in new cases, especially for tensor and symmetric ranks.

## Key findings

- Monic rank is finite and ≥ usual rank.
- Algorithmic method to determine maximal monic rank.
- Confirmed cases where maximal rank equals maximal monic rank.

## Abstract

We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone $X$. We show that the monic rank is finite and greater than or equal to the usual $X$-rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to B. Shapiro which states that a binary form of degree $d\cdot e$ is the sum of $d$ $d$-th powers of forms of degree $e$. Furthermore, in the case where $X$ is the cone of highest weight vectors in an irreducible representation---this includes the well-known cases of tensor rank and symmetric rank---we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances.