實係數多項式函數是一門廣泛應用於數學上的基礎知識，並且多項式函數除了在數學上很重要以外在很多領域中，諸如物理、金融、天文、機械工程......等等都是不可或缺的一環，然而在這些領域中利用函數來建立特定情況的模型後，函數的根通常具有不同的重要意義，因此如何找到這些函數的根成為了一項重要的課題。 關於勘根的方法，從古代到現在經歷了漫長的發展，最早可追溯到公元前2000年時古巴比倫的數學家就能解出二次函數的根，之後演變成如今的一元二次方程式的公式解，之後的數學家們更進一步努力想要找出更高次函數的解，於是衍生出了卡爾達諾公式(Cardano formula)來解決三次函數的根和費拉里（Lodovico Ferrari）的降階法來解決四次函數的根。在18世紀末到19世紀初時阿貝爾-魯菲尼定理(Abel–Ruffini theorem)證明五次及更高次的多項式方程式沒有一般的求根公式，因此為了找到高次多項式的根各種勘根方式先後被提出，有別於勘根定理只能粗淺判斷區間中的根為奇數個還是偶數個，本文將帶領大家一起欣賞由四位數學家橫跨數個世紀一步步完善的符號律如何精確找到區間中的實根數量。 本文將帶領大家一起探討符號律是如何從勒內·笛卡兒(René Descartes)開始一路到雅克·夏爾·法蘭斯瓦·史特姆(Jacques Charles François Sturm)利用各種方法找到隱藏於函數之中的一串數列，並利用這串數列的正負變化來找出區間中的實根數量，從一開始只能粗淺知曉正根的大致數量，到後來弗朗索瓦·布丹(François Budan)和約瑟夫·傅里葉(Joseph Fourier)兩位數學家繼承笛卡爾的想法將勘根的範圍從只能搜索正根到可以搜索任異區間，最後引出史特姆利用輾轉相除法找到可以計算區間中實根數量的方法。透過這幾位數學家的努力完成了對任意實係數多項式函數的勘根，讓我們面對高次函數時，也可以有方法利用符號律來定位出根的位置，並且在電腦輔助計算的情況下我們可以透過分割實數的方式來找到實根。 隨著電腦運算能力的進步，除了利用史特姆定理以外，從19世紀開始也陸續出現許多不同的數值方法，例如，二分法、牛頓法、割線法、QR分解法、Durand-Kerner法......等等，各種用以應對不同情況的數值方法，使用這些不同的工具也讓我們在各領域中建造出函數後都能更容易地找出其中的根。 本文共分六節，第一節討論笛卡爾符號律，第二節討論Budan-Fourier Theorem，第三節討論Sturm Theorem，第四節為三個符號律實際應用的例子，第五節是對本文證明中所使用的微積分工具作一基本的證明，第六節是計算時所使用的python程式；其中笛卡爾符號律為笛卡爾在1637年發表的《幾何學》（La Géométrie）中提出的、Budan-Fourier Theorem為布丹在 1807 年先提出了初步的理論再由傅里葉在1820年進一步改進而成的、Sturm Theorem 為史特姆在1829年發表的。

Polynomial functions with real coefficients form a fundamental topic in mathematics, with broad applications beyond mathematics itself. These functions play an indispensable role in various fields such as physics, finance, astronomy, and mechanical engineering. In these disciplines, mathematical functions are often used to model specific situations, and the roots of these functions carry different important meanings. As a result, finding the roots of polynomial functions has become a crucial topic of study. The methods for root-finding have undergone a long historical development. The earliest records date back to around 2000 BCE when Babylonian mathematicians were already capable of solving quadratic equations. This later evolved into the quadratic formula used today. Mathematicians then strived to find solutions for higher-degree polynomial equations, leading to the development of the Cardano formula for solving cubic equations and Ferrari’s method for reducing quartic equations. In the late 18th and early 19th centuries, the Abel-Ruffini theorem proved that there is no general formula for solving polynomial equations of degree five or higher. Consequently, various numerical and analytical root-finding methods were developed. Unlike root-finding theorems that only determine whether a given interval contains an odd or even number of roots, this article will explore how the rule of signs, refined over centuries by four mathematicians, allows for an exact determination of the number of real roots within an interval. This article will guide readers through the development of the rule of signs, from its origins with René Descartes to its refinement by Jacques Charles François Sturm. We will examine how these mathematicians discovered a sequence of numbers hidden within a polynomial function and how the variations in sign within this sequence can reveal the number of real roots in an interval. Initially, Descartes rule of signs could only estimate the number of positive roots, but later, François Budan and Joseph Fourier expanded its applicability to arbitrary intervals. Finally, Sturm utilized the Euclidean algorithm to develop a method for precisely counting real roots within a given interval. Thanks to the contributions of these mathematicians, root-finding for any polynomial with real coefficients became a systematic process. Even when dealing with high-degree polynomials, we can now apply the rule of signs to locate roots efficiently. Furthermore, with computer-assisted calculations, we can use numerical methods to further refine the root-finding process. With the advancement of computational power, various numerical methods have been developed since the 19th century alongside Sturm’s theorem. These include the bisection method, Newton’s method, the secant method, QR decomposition, the Durand-Kerner method, and others each suited to different cases. By utilizing these tools, we can efficiently determine the roots of polynomial equations in diverse fields after constructing the corresponding functions. This article is divided into six sections: 1. Descartes Rule of Signs 2. The Budan-Fourier Theorem 3. Sturm Theorem 4. Practical Applications of the Three Root-Finding Rules 5. Fundamental Calculus Proofs Used in This Paper 6. Python Code Used for Computation Descartes rule of signs was introduced by René Descartes in his 1637 work La Géométrie. The Budan-Fourier theorem was initially proposed by François Budan in 1807 and further refined by Joseph Fourier in 1820. Sturm’s theorem was published by Jacques Charles François Sturm in 1829.

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