English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In 2016 J. Koenigsmann refined a celebrated theorem of J. Robinson by provingthat $\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$, i.e., thereis a polynomial $P(t,x_1,\ldots,x_{n})\in\mathbb Z[t,x_1,\ldots,x_{n}]$ suchthat for any rational number $t$ we have $$t\not\in\mathbb Z\iff \existsx_1\cdots\exists x_{n}[P(t,x_1,\ldots,x_{n})=0]$$ where variables range over$\mathbb Q$, equivalently $$t\in\mathbb Z\iff \forall x_1\cdots\forallx_{n}[P(t,x_1,\ldots,x_{n})\not=0].$$ In this paper we prove that we may take$n=32$. Combining this with a result of Z.-W. Sun, we show that there is noalgorithm to decide for any $f(x_1,\ldots,x_{41})\in\mathbbZ[x_1,\ldots,x_{41}]$ whether $$\forall x_1\cdots\forall x_9\existsy_1\cdots\exists y_{32}[f(x_1,\ldots,x_9,y_1,\ldots,y_{32})=0],$$ wherevariables range over $\mathbb Q$.

$\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$ with 32 unknowns