This may be quite a broad question, but lately I've been wondering about the history behind polynomials. Nowadays these are pretty much the simplest kind of functions to work with, but I'd like to know how this came to be.

I'd also like to know who was/were (probably) the first to translate practical problems into solving polynomial equations, not using geometric methods like the greeks, but something closer to the modern approach.

In this answer, Conifold mentions Indian scientists knew some Taylor series already around the fifth century, so the importance of polynomial equations was already known by then. On the other hand, I believe greeks didn't know about it (I know this assertion is too broad).

Summarising:

What led to the realisation that polynomials were so important and when did it happen?

and

What philosophical implications did it have?

  • 1
    Madhava series is from 14th century, and Aryabhata in the 5th did not go beyond the second order. Diophantus had a much better grasp on polynomials c. 250 AD. – Conifold Jul 8 '15 at 3:37
up vote 11 down vote accepted

Although many problems that we now reduce to polynomial equations were solved since time immemorial early occurences are coached in verbal and/or geometric terms, and polynomials are not treated as separate items. For early occurences of geometric problems that lead (today) to quadratic equations see The origin of quadratic equation in actual practice. The equations proper, let alone polynomials themselves, do appear only relatively late.

The first breakthrough was made by Diophantus in Arithmetica (c.250 AD), who introduced notation for an indeterminate $\varsigma$, the first $x$, and its powers up to sixth, $\Delta^{\upsilon}$ is $\varsigma$ squared, $K^{\upsilon}$ cubed, etc. So for the first time it became possible to write down polynomials, albeit only up to degree six. He even had symbols for constant terms and reciprocals of powers. Diophantus was also first to convert text problems into polynomial equations, and use some rudimentary (verbal) algebra to solve them. However, Diophantus's "polynomials" are not yet separate items either, they only appear as sides in equations. This is analogous to Babylonian and Hellenistic use of zero symbol as a placeholder. Here is $3x^2+1-(10x^3+2x)=4$ in Diophantus's notation: $\Delta^{\upsilon}\gamma\mathring{M}\alpha\pitchfork K^{\upsilon}\iota\varsigma\beta\iota^{\sigma}\mathring{M}\delta$. See Hettle's The Symbolic and Mathematical Influence of Diophantus's Arithmetica for details and notation.

The next great step was made by medieval Islamic mathematicians. The familiar rules of algebra were formalized by Al-Khowarizmi (ca. 800-847 AD), who also derived first algebraic solutions to linear and quadratic equations (before him they were either given as prescriptions, or geometrically concealed as in Euclid). Perhaps the first person to conceive of general polynomials is Al-Karaji (953-1029 AD), who can be called their "discoverer". He realized that the series of powers (and their reciprocals) extends indefinitely. Correspondingly, he extended Diophantus's notation to write polynomials of arbitrary degree, and gave rules for their addition, subtraction and multiplication.

But polynomials truly come to shine in the work of Al-Samawal (1130-1180) The Shining Book on Calculation. He dispensed with verbal and semi-verbal notation, and used tables of coefficients to write and perform calculations with polynomials, including those with negative powers (Laurent polynomials). This greatly simplified all algebraic calculations with them because "laws of exponents" are applied automatically. And he gives the first polynomial division algorithm, the grandfather of modern long and synthetic division. He also noted the analogy between his way of writing polynomials and the decimal positional notation, and transferred his algorithms to decimal numbers by replacing the variable with $10$. This was the first mathematical justification of a positional division algorithm. See Islamic Mathematics and Katz's History of Mathematics.

Unfortunately, Renaissence Europe did not absorb Al-Samawal's innovations, and instead proceeded through incremental improvement of notation. In particular, the notation used by del Ferro, Tartaglia and even Cardano to solve the cubic in 1500s was largely verbal, and inferior to Al-Samawal. Even so, in Ars Magna (1545) Cardano introduced the technique of substitutions that not only solved the cubic and quartic, but became indispensable in polynomial algebra later. See Why is "Cardano's Formula" (wrongly) attributed to him?

Viète's Isagoge (1591) introduced modern style symbolic notation and algebraic manipulation rules Viète still uses words for powers, these were symbolized by Descartes, but they are attached to variables. In particular the use of letters for parameters allowed general consideration of polynomials rather than example by example. And philosophically, in Viète's works we for the first time encounter a systematic use of the method where problems are converted to equations, and then solved algebraically. See Viète's Relevance and his Connection to Euler, and references therein.

This method was further sharpened by Descartes's analytic geometry in La Géométrie (1637). He connected it to the classical method of analysis and synthesis described by Pappus, but with conversion to algebraic equations in the middle. This not only streamlined solutions of many classical problems, but also covered many new ones, expressible by higher order equations. Moreover, Descartes considers polynomials in two variables, which represented algebraic curves in analytic geometry, and this is where algebraic geometry takes its root. Descartes's formalization of construction methods, and classification of problems based on their algebraic representation, led to techniques required for impossibility proofs, starting with Gregory's (unsuccessful) attempt to prove algebraic unsolvability of quadrature (1667). See Crippa's Impossibility Results: from Geometry to Analysis.

Polynomial equations are among the primary objects of mathematics since the beginning. Babylonians knew how to solve quadratic equations though they did not use modern notation of course. The theory of conic sections is the study of quadratic equations in 2 variables, and it was performed by Apollonius geometrically. (At that time algebraic notation did not exist, and the Greeks did not think of the conic sections as polynomial equations). Polynomial equations of higher degree and in many variables were first systematically studied by Diophantus, over the field of rational numbers, so in a sense Apollonius and Diophantus can be considered as creators of algebraic geometry (over the reals and over the rationals, respectively).

Muslim scientists continued the study of polynomials during the "Dark Age" in Europe.

It was natural to search of a general formula for solution of the polynomial equations in one variable of degree higher than 2. This was accomplished in XVI century (Tartaglia, Cardano, Ferrari). Modern notation for polynomials was introduced by Vieta. Finally Descartes invented his coordinate method and connected the geometric approach of the Greeks with the algebraic approach. So Descartes was probably the first who studied systems of polynomial equations as we know them now.

So by the time of Newton, study of polynomial equations was very well established. Newton himself contributed to this by classifying the real algebraic curves of degree 3 (for degree 2 this was done by Apollonius, and in algebraic notation by Descartes).

Let me add that polynomials are not just the "simplest functions" but remain one of the principal objects of study in mathematics nowadays. The area called algebraic geometry is a central part of mathematics. It studies polynomial equatons.

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