Hadamard products of hypersurfaces (Q2153793)

This paper investigates Hadamard products and hypersurfaces. Let \(P = [a_0: a_1 : \cdots : a_n]\) and \(Q = [b_0: b_1: \cdots : b_n]\) be two projective points in \(\mathbb P^n\). If \(a_ib_i \not = 0\) for some \(i\), then we define the Hadamard product of the points to be \[ P \star Q = [a_0b_0:a_1b_1: \cdots: a_nb_n]. \] If \(a_ib_i = 0\) for \(1 \leq i \leq n\), then \(P \star Q\) is not defined. We extend this to two projective varieties \(V\) and \(W\) in \(\mathbb P^n\): \[ V \star W = \overline{\{P \star Q: P \in V, Q \in W, P \star Q \, \, \text{is defined}\}} \] where the closure is taken in the Zariski topology. We also define the \(r^{\mathrm{th}}\) Hadamard product of a variety \(V\) as \[ V^{\star r} = V \star V^{\star(r-1)} \] where \(V^{\star 0} = [1: \cdots : 1]\). To set some notation, let \(H_i\) denote the hyperplane \(x_i = 0\) in \(\mathbb P^n\) and set \[ \Delta_i = \bigcup_{0 \leq j_1 \leq \cdots \leq j_{n-1} \leq n} H_{j_1} \cap \cdots \cap H_{j_{n-i}}. \] For a vector of nonnegative integers \(I = (i_0, \ldots, i_n)\) and \(P = [p_0 : \cdots : p_n]\) in \(\mathbb P^n\) we set: \begin{itemize} \item \(X^I = x_0^{i_0} x_1^{i_1} \cdots x_n^{i_n} \in \mathbb C[x_0, \ldots, x_n]\); \item \(|I| = i_0 + \cdots + i_n\); \item \(P^I = p_0^{i_0}p_1^{i_1} \cdots p_n^{i_n}\); \item if \(P \in \mathbb P^n \setminus \Delta_{n-1}\), then \(\frac{1}{P}\) is the point \(\left[ \frac{1}{p_0} : \frac{1}{p_1} : \cdots : \frac{1}{p_n} \right]\). \end{itemize} Given a homogeneous polynomial \(f = \sum_{|I| = d} a_IX^I \in \mathbb C[x_0, x_1, \ldots, x_n]\) of degree \(d\) and \(P \in \mathbb P^n \setminus \Delta_{n-1}\), the authors define the Hadamard transformation of \(f\) by \(P\) to be the polynomial \[ f^{\star P} = \sum_{|I| = d} \frac{a_I}{P^I}X^I. \] The main result of Section 3 is: Theorem. Let \(V \subset \mathbb P^n\) be a variety and \(P \in \mathbb P^n \setminus \Delta_{n-1}\). If \(f_1, \ldots, f_s \subset \mathbb C[x_0, \ldots, x_n]\) is a generating set for \(I(V)\), then \(f_1^{\star P}, \ldots, f_s^{\star P}\) is a generating set for \(I(P \star V)\). Moreover, if \(f_1, \ldots, f_s\) is a Gröbner basis for \(I(V)\), then \(f_1^{\star P}, \ldots, f_s^{\star P}\) is a Gröbner basis \(I(P \star V)\). This Theorem is then used to investigate Hadamard products of hypersurfaces. Note that an irreducible hypersurface is said to be a binomial hypersurface if it has defining equation of the form \[ \alpha_1X^{I_1} - \alpha_2 X^{I_2} = 0. \] The main results of Section 4 study binomial hypersurfaces with similar defining equations. In particular, the authors prove the following two propositions. Proposition. If \(C, D \subset \mathbb P^n\) are the binomial hypersurfaces \[ C = Z(\alpha_1X^{I_1} - \alpha_2 X^{I_2})\text{ and }D = Z(\beta_1X^{I_1} - \beta_2 X^{I_2}) \] then \[ C \star D = Z(\alpha_1\beta_1 X^{I_1} - \alpha_2\beta_2 X^{I_2}). \] Proposition. Let \(C, D\) be irreducible hypersurfaces not contained in \(\Delta_{n-1}\). If \(C \star D\) is a hypersurface, then \[ C = Z(\alpha_1X^{I_1} - \alpha_2 X^{I_2})\text{ and }D=Z(\beta_1X^{I_1} - \beta_2 X^{I_2}). \] Moreover, \(C \star D = Z(\alpha_1\beta_1 X^{I_1} - \alpha_2\beta_2 X^{I_2})\). The remainder of the paper is dedicated to studying Hadamard powers of hypersurfaces and involves similar results as above. An extra condition is introduced in the case of general binomial hypersurfaces. Namely, the authors consider binomial hypersurfaces \(C \subset \mathbb P^n\) of type \((t, \epsilon)\) meaning \(\alpha_1 = 1\) and \(\alpha_2 = \xi^{\epsilon}\) where \(\xi\) is a primitive \((t-1)\)-th root of unity and \(1 \leq \epsilon \leq t-1\). By considering such a condition, the authors generalize a result of [\textit{N. Friedenberg} et al., Fields Inst. Commun. 80, 133--157 (2017; Zbl 1390.14157)] which says that if \(V \subset \mathbb P^n\) is a projective variety generated by an ideal with a minimal set of generators of the type \(X^{\alpha} - X^{\beta}\) with \(\alpha, \beta \in \mathbb N^{n+1}\), then \(V^{\star 2} = V\). The authors' generalization requires the following definition: Definition. A binomial variety \(C \subset \mathbb P^n\) is of type \([(t_1, \epsilon_1), \ldots, (t_s, \epsilon_s)]\) if the ideal of \(C\) is generated by \[ X^{I_{1,1}} - \xi_1^{\epsilon_1}X^{I_1,2}, \ldots, X^{I_{s,1}} - \xi_s^{\epsilon_s}X^{I_s,2} \] where \(\xi_i\) is a primitive \((t_1-1)\)-th root of unity and \(1 \leq \epsilon_i \le t_i-1\) for \(i = 1, \ldots, s\). The main theorem of the paper that generalizes the work mentioned above says: Theorem. Let \(C \subset \mathbb P^n\) be a binomial variety of type \([(t_1, \epsilon_1), \ldots, (t_s, \epsilon_s)]\) with \[ \mathrm{lcm}\left( \frac{t_1}{\gcd(t_1, \epsilon_1)}, \ldots, \frac{t_s}{\gcd(t_s, \epsilon_s)} \right) = t-1. \] Then \(C^{\star t} = C\) and \(t\) is the minimal exponent for which this equality holds.