Tomasz Kowalczyk

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20242026

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math.AG2026

A note on the real Jacobian conjecture in degree 7

Tomasz Kowalczyk

Let $(p, q)$ be a Jacobian pair. We show that the real Jacobian conjecture holds if the degree of $p$ is 7 and the highest degree homogenous part is of the form $α x^{7} + β x^{6} y$ for $…

math.AG2025

Sums of squares of regular functions on rational surfaces

Tomasz Kowalczyk

We study the sums of squares on cylinders of the form $X \times A_{K}$ for a (weakly) factorial curve $C$ . We prove the equality of the Pythagoras numbers of the ring of reg…

math.AG2024

On higher Pythagoras numbers of polynomial rings

Tomasz Kowalczyk, Julian Vill

We show that the higher Pythagoras numbers for the polynomial ring are infinite $p_{2 s} (K [x_{1}, x_{2}, \dots, x_{n}]) = \infty$ provided that $K$ is a formally real field, $n \geq 2$ and $s\ge…