r-fat polynomials are a natural generalization of scattered polynomials. They define linear sets of the projective line $\mathrm{PG}(1,q^n)$ of rank n with r points of weight larger than one. Using techniques on algebraic curves and function fields, we obtain numerical bounds for r and the non-existence of exceptional r-fat polynomials with $r>0$. We completely determine the possible values of r when considering linearized polynomials over ${\mathbb{F}}_{q^4}$ and we also provide one family of 1-fat polynomials in $\mathrm{PG}(1,q^5)$. Furthermore, we investigate LP-polynomials (i.e. polynomials of type $f(x)=x+\delta x^{q^{2s}}\in {\mathbb{F}}_{q^n}[x]$, $\gcd (n,s)=1$), determining the spectrum of values r for which such polynomials are r-fat.

r -fat 多项式是分散多项式的自然推广。他们定义了投影线的线性集$\mathrm{PG}(1,q^n)$秩为n且r个权重大于 1 的点。使用代数曲线和函数域的技术，我们获得了r的数值界限和不存在异常r -fat 多项式$r>0$. 当考虑线性化多项式时，我们完全确定了r的可能值${\mathbb{F}}_{q^4}$我们还提供了一组 1-fat 多项式$\mathrm{PG}(1,q^5)$. 此外，我们研究 LP 多项式（即多项式类型$F(X)=X+\delta X^{q^{2s}}\in {\mathbb{F}}_{q^n}[X]$,$\gcd (n,s)=1$)，确定这些多项式为r -fat的值r的频谱。