給定兩個定義在 R^n 上的二次函數 f(x) = x^⊤Ax + 2a^⊤x + a_0 和 g(x) = x^⊤Bx + 2b^⊤x + b_0，本論文主要考慮它們的聯合值域 R(f, g) = {(f(x), g(x)) | x ∈ R^n}。雖然這是一個重要的問題，但由文獻中可知，我們對聯合值域 R(f, g) 的幾何形狀仍所知甚少。目前僅對於聯合值域 R(f, g) 的凸性，從 1941 年 Dines 的工作 [5] 到 2016 年 Bazán 和 Opazo [6] 和 2022 年 Nguyen 等人的工作 [8]，最終才給出了判斷 R(f, g) 凸性的充分必要條件。本論文我們除了回顧聯合值域 R(f, g) 凸性的條件之外，特別進一步聚焦在聯合值域 R(f, g) 的可能形狀。儘管這是一個很遠大的目標，但我們成功地踏出第一步在 R^2 和 R^3 上對一組仿射函數 f 和二次函數 g 的所有可能之聯合值域 R(f, g) 形狀進行了分類；同時也在 R^2 和 R^3 上兩個齊次二次函數的一部分特殊情況有了部分進展。論文中我們特別設計了許多表格，以總結聯合值域 R(f, g) 的各種類型以及與之相關的函數 f 和 g 所需要滿足的形式，作為方便之後研究查閱和參考之用。此外，我們提供了許多具體的數值例證，並通過這些例證和圖表來加深對理論結果的理解，為研究二次聯合值域圖形分類的複雜性做出初步的貢獻。

Given two quadratic functions f(x) = x^⊤Ax + 2a^⊤x + a_0 and g(x) = x^⊤Bx + 2b^⊤x + b_0, this paper mainly considers their joint numerical range R(f, g) = {(f(x), g(x)) | x ∈ R^n}. Although this is an important question, we know very little about the geometry of the joint numerical range R(f, g) from the literature. At present, only for the convexity of the joint numerical range R(f, g), from the work of Dines [5] in 1941 to the work of Bazán and Opazo [6] in 2016 and the work of Nguyen et al. [8] in 2022, necessary and sufficient conditions for checking the convexity of R(f, g) are finally given. In this paper, in addition to reviewing the conditions for the convexity of the joint numerical range R(f, g), we further explore the possible shapes of the joint numerical range R(f, g). Although this is an ambitious goal, we succeeded in taking the first step to classify all possible joint domain R(f, g) shapes for a set of affine functions f and quadratic functions g on R^2 and R^3; at the same time, progresses in some special cases of two homogeneous quadratic functions on R^2 and R^3 are also obtained. We also designed many tables to summarize the various types of the joint numerical range R(f, g) and the related conditions for functions f and g to be satisfied. This serves as a convenient reference and tool for later research. In addition, we provide many specific numerical examples and use these examples and graphs to deepen the understanding of theoretical results, making initial contributions to classify the possible graphs of the quadratic joint numerical range.

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