当前位置: X-MOL 学术Eur. J. Comb. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
On Rado conditions for nonlinear Diophantine equations
European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2020-12-17 , DOI: 10.1016/j.ejc.2020.103277
Jordan Mitchell Barrett , Martino Lupini , Joel Moreira

Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These conditions are inspired by Rado’s characterization of partition regular linear homogeneous equations. We conjecture that these conditions are also sufficient for partition regularity, at least for equations whose corresponding monovariate polynomial is linear. This would provide a natural generalization of Rado’s theorem.

We verify that such a conjecture holds for the equations x2xy+ax+by+cz=0 and x2y2+ax+by+cz=0 for a,b,cZ such that abc=0 or a+b+c=0. To deal with these equations, we establish new results concerning the partition regularity of polynomial configurations in Z such as x,x+y,xy+x+y, building on the recent result on the partition regularity of x,x+y,xy.



中文翻译:

关于非线性Diophantine方程的Rado条件

在Di Nasso和Luperi Baglini的先前工作的基础上,我们提供了Diophantine方程成为分区正规的一般必要条件。这些条件是受Rado对分区正则线性齐次方程的刻画所启发。我们推测,这些条件也足以满足分配规则性,至少对于其单变量多项式为线性的方程式而言已经足够。这将提供拉多定理的自然概括。

我们验证了这样的猜想对等式成立 X2-Xÿ+一种X+bÿ+Cž=0X2-ÿ2+一种X+bÿ+Cž=0 对于 一种bCž 这样 一种bC=0 要么 一种+b+C=0。为了处理这些方程,我们建立了关于多项式配置的分区正则性的新结果。žXX+ÿXÿ+X+ÿ,基于最近的分区规则性结果 XX+ÿXÿ

更新日期:2020-12-17
down
wechat
bug