This article deals with Puiseux expansions and non isolated points in algebraic varieties.
By means of an equidimensional decomposition of the algebraic variety \(V(f)\), one can describe the set of complex zeroes of a finite family of multivariate polynomials with rational coefficients.
In this paper, the problem of deciding whether a common solution to a multivariate polynomial equation system is isolated or not is considered.
The motivation to this problem comes from the fact that for certain families of polynomial systems, in the symbolic framework, larger sets of points representing the equidimensional components of a variety can be computed with better complexities. Somehow, no algorithm discarding extra points within the same complexity is known.
This problem motivates the search for new symbolic tools. To this end, the first step is to decide algorithmically whether the point on the variety is isolated or not.
In this work, two cases are considered. The first case is a system of two bi-variate polynomials.
In the case of 1-dimensional varieties, one has a theorem ensuring that a point \(\xi\) on the variety is not isolated.
These conditions are such that the initial part of a given truncated Puiseux series vector centered at that point \(\xi\) coincides with the initial part of a Puiseux series expansion of a parametrization of a curve in the 1-dimensional variety containing this point \(\xi\) .
The second case is a consideration of an arbitrary number of polynomials. In the case of non-isolated points, one has the following statement:
Theorem. Let \(f=(f_{1}, \dots, f_{m})\) be a system of \(n\)-variate complex polynomials. Let \(\xi=(\xi_{1}, \dots, \xi_{n})\in \mathbb{C}^{n}\) be a zero of this system. Let \(\gamma_{i},L \in\mathbb{Q}\) with \(i\in\{0,\dots, N\}\) such that \(\gamma_{0}<\dots<\gamma_{N}\leq L\) and
\[ \Theta=\left(t, \sum a_{i2}(t-\xi_{1})^{\gamma_{i}},\dots, \sum_{i=0}^{N}a_{in}(t-\xi_{1})^{\gamma_{i}}\right) \]
be a Puiseux series vector with coefficients in \(\mathbb{C}\) centered at \(\xi_{1}\) such that \(a_{0l}=\xi_{l}\) for all \(2\leq l\leq n \) and \(\mathrm{ord}_{(t-\xi_{1})}(f_{j}(\Theta))>L\) for all \(1\leq j\leq m\). Let \(e(f)\) be the Noether exponent of \(\big \langle f_{1}, \dots, f_{m}\big \rangle\). If \(L\geq e(f)\) then there exists an irreducible component \(W\) of \(V(f)\) with free variable \(X_{1}\) such that \(\xi \in W\).
An example is given showing that the bound \(L\geq e(f)\) in this theorem is sharp.
In the case of 1-varieties:
Theorem. Let \(f=(f_{1}, \dots, f_{m})\) be a system of n-variate complex polynomials such that \(\dim(V(f))\leq 1\) and \(\xi=(\xi_{1}, \dots, \xi_{n})\in \mathbb{C}^{n}\) be a zero of this system.
Let \(\gamma_{i},L \in\mathbb{Q}\) with \(i\in\{0,\dots, N\}\) such that \(\gamma_{0}<\dots<\gamma_{N}\leq L\) and
\[ \Theta=\left(t, \sum a_{i2}(t-\xi_{1})^{\gamma_{i}},\dots, \sum_{i=0}^{N} a_{in}(t-\xi_{1})^{\gamma_{i}}\right) \]
be a Puiseux series vector with coefficients in \(\mathbb{C}\) centered at \(\xi_{1}\) such that \(a_{0l}=\xi_{l}\) for all \(2\leq l\leq n\) and \(\mathrm{ord}_{(t-\xi_{1})}(f_{j}(\Theta))>L\) for all \(1\leq j\leq m\).
Let \(e(f)\) be the Noether exponent of \(\big \langle f_{1}, \dots, f_{m}\big \rangle\). If \(L\geq e(f)\mathrm{deg}(V(f))\), there exists a curve \(W\) in \(V(f)\) with free variable \(X_{1}\) such that \(\xi \in W\) and there is a parametrization of \(W\) whose initial terms are \(\Theta_{M}:=\left(t,\sum_{i=0}^{M}a_{i2}(t-\xi_{1})^{\gamma_{i}},\dots, \sum_{i=0}^{M}a_{in}(t-\xi_{1})^{\gamma_{i}}\right)\) where \[ M=\max\left\{i\in \{0,\dots N\}| \gamma_{i}\leq \frac{L}{e(f)\mathrm{deg}(V(f))}\right\}. \]
An example is given showing that the precision order is sharp for certain choices of the parameters.