A note on Schmidt's subspace type theorems for hypersurfaces in subgeneral position (Q6540613)

Before discussing some results of this paper, we introduce some notation. Let \(k\) be a number field and \(M_k\) its set of places. For a place \(v\) of \(k\) choose an absolute value \(|\cdot |_v\) such that if \(v\) lies above \(p\in\{\infty\}\cup\{\)primes\(\}\) then the restriction of \(|\cdot |_v\) to \(\mathbb{Q}\) is \(|\cdot |_p\). Subsequently, define \(\|\cdot \|_v:=|\cdot |_v^{[k_v:\mathbb{Q}_p]/[k:\mathbb{Q}]}\). Define the logarithmic height of \(\textbf{x}=(x_0,\ldots ,x_n)\in\mathbb{P}^n(k)\) by \(h(\textbf{x}):=\sum_{v\in M_k}\log\max_{0\leq i\leq n}\| x_i\|_v\). For a hypersurface \(D\) of \(\mathbb{P}^n\) of degree \(d\) defined over \(k\), say by \(Q=0\) where \(Q\in k[x_0,\ldots ,x_n]\) is a homogeneous polynomial of degree \(d\) define\N\[\N\lambda_{D,v}(\textbf{x}):=\log\left( \Big(\max_{0\leq i\leq n}\|x_i\|_v\Big)^d\|Q\|_v/\| Q(\textbf{x})\|_v\right)\ \ (v\in M_k),\N\]\Nwhere \(\|Q\|_v\) is the maximum of the \(\|\cdot\|_v\)-values of the coefficients of \(Q\), and\N\[\Nm_{D,S}(\textbf{x}):=\sum_{v\in S}\lambda_{D,v}(\textbf{x})\N\]\Nfor \(\textbf{x}\in\mathbb{P}^n(k)\), where \(S\) is a finite subset of \(M_k\).\N\NWe say that hypersurfaces \(D_1,\ldots ,D_q\) of \(\mathbb{P}^n\) are in \(m\)-subgeneral position, with \(m\geq n\), if any \(m+1\) among them have empty intersection. Building further on work of \textit{P. Corvaja} and \textit{U. Zannier} [Am. J. Math. 126, No. 5, 1033--1055 (2004; Zbl 1125.11022); addendum ibid. 128, No. 4, 1057--1066 (2006)] and \textit{J.-H. Evertse} and \textit{R. G. Ferretti} [Dev. Math. 16, 175--198 (2008; Zbl 1153.11032)], \textit{Si Duc Quang} [Int. J. Number Theory 15, No. 4, 775--788 (2019; Zbl 1452.11083); Pac. J. Math. 318, No. 1, 153--188 (2022; Zbl 1501.11073)] proved the following version of Schmidt's Subspace Theorem: \N\NTheorem. Let \(S\) be a finite set of places of \(k\) and \(D_1,\ldots ,D_q\) hypersurfaces of \(\mathbb{P}^n\) defined over \(k\) of degrees \(d_1,\ldots ,d_q\), respectively, in \(m\)-subgeneral position, where \(q\geq m\geq n\). Then for every \(\varepsilon >0\) there exists a proper Zariski-closed subset \(Z\) of \(\mathbb{P}^n\) such that\N\[\N\sum_{j=1}^q d_j^{-1}m_{D_j,S}(\textbf{x})\leq \big( (m-n+1)(n+1)+\varepsilon\big)h(\textbf{x})\N\]\Nholds for all \(\textbf{x}\in\mathbb{P}^n(k)\setminus Z\).\N\NTo obtain this result, Quang introduced his `replacing hypersurfaces technique.'\N\NThe authors prove a similar result, but with \((m-n+1)(n+1)\) replaced by \(c(m,n,t)\), which is defined as follows. Let \(t\) be the smallest integer such that for any \(t+1\) hypersurfaces \(D_{j_1},\ldots ,D_{j_{t+1}}\in\{ D_1,\ldots D_q\}\) one has \(\dim D_{j_1}\cap\cdots\cap D_{j_{t+1}}\leq n-2\). Then\N\[\Nc(t,m,n):=(m-n+2)(n+1)/2\ \ \text{if } 1\leq t\leq (m-n+2)/2,\N\]\N\[\Nc(t,m,n):=\min_{\theta\in [0,t]} t-\theta +(n+1)\max\Big( \frac{m}{n}, \theta +\frac{t-\theta}{t}(m-n-t+2)\Big)\N\]\Nif \((m-n+2)/2 <t\leq m-n+1\).\N\NFor instance, if \(D_1,\ldots ,D_q\) are distinct and irreducible then \(t=1\) and \(c(m,n,t)=(m-n+2)(n+1)/2\) which improves upon the term \((m-n+1)(n+1)\) in Quang's result.\N\NThe authors use Quang's ideas, but add some combinatorial innovations.