What is algebraic geometry, and what is the need for a new book on it?
First, we take up the question of what is Algebraic Geometry. Long ago, to
a major extent in my father's time, and to a lesser extent in my own time,
in high-school and college we learned the two subjects of analytic geometry
and theory of equations. Analytic geometry consists of studying geometric
figures by means of algebraic equations. Theory of equations, or high school
algebra, was manipulative in nature and dealt with simplifying expressions,
factoring polynomials, making substitutions, and solving equations. These
two subjects were later synthesized into and started being collectively called
algebraic geometry. Thus, algebraic geometry, at least in its classical form, is
an amalgamation of analytic geometry and the theory of equations.
But, in the last fifty years, algebraic geometry, as such, became more and
more abstract, and its original two incarnations, mentioned above, gradu-
ally vanished from the curriculum. Indeed, analytic geometry first became a
chapter, and then a paragraph, and finally only a footnote in books on cal-
culus. Likewise, its sister discipline of trigonometry, with all the proving of
identities, began to be downplayed. Doing all these manipulations was cer-
tainly helpful in enhancing the skills needed for solving intricate problems.
Similarly, studying subjects like analytic geometry and trigonometry was very
useful in developing geometric intuition.
Now, during the last ten years or so, with the advent of the high-speed com-
puter, the need for the manipulative aspects of algebra and algebraic geometry
is suddenly being felt in the scientific and engineering community. The grow-
ing and dominating abstractions of algebraic geometry notwithstanding, my
approach to it remained elementary, manipulative, and algorithmic. In my
1970 poem, "Polynomials and Power Series," and my 1976 article on "His-
torical Ramblings," I lamented the passing of the concrete attitude and made
a plea for its rejuvenation. Thus, it is with great pleasure that I see the recent
rise of the algorithmic trend, albeit at the hands of the engineers, and I am
happy for the company of their kindred souls.
In this book on algebraic geometry, which is based on my recent lectures
to an engineering audience, I am simply resurrecting the concrete and ancient
methods of Shreedharacharya (500 A.D.), Bhaskaracharya (1150 A.D.), New-
ton (1660), Sylvester (1840), Salmon (1852), Max Noether (1870), Kronecker
(1882), Cayley (1887), and so on.