Merge pull request #187 from zcash/book-fixes

Book fixes

book/src/background/curves.md

@@ -249,4 +249,4 @@ framework used in the Internet Draft makes use of several functions:
 Reference: https://eprint.iacr.org/2019/403.pdf
 
 ## References
-[^complete-formulae]: Renes, J., Costello, C., & Batina, L. (2016, May). "Complete addition formulas for prime order elliptic curves." In Annual International Conference on the Theory and Applications of Cryptographic Techniques (pp. 403-428). Springer, Berlin, Heidelberg. https://eprint.iacr.org/2015/1060.pdf
+[^complete-formulae]: [Renes, J., Costello, C., & Batina, L. (2016, May). "Complete addition formulas for prime order elliptic curves." In Annual International Conference on the Theory and Applications of Cryptographic Techniques (pp. 403-428). Springer, Berlin, Heidelberg.](https://eprint.iacr.org/2015/1060)

book/src/background/fields.md

@@ -294,4 +294,4 @@ Important notes:
   of unity). There is a two-to-one mapping between the elements and their squares.
 
 ## References
-[^chinese-remainder]: Friedman, R. (n.d.) "Cyclic Groups and Elementary Number Theory II" (p. 5). http://www.math.columbia.edu/~rf/numbertheory2.pdf
+[^chinese-remainder]: [Friedman, R. (n.d.) "Cyclic Groups and Elementary Number Theory II" (p. 5).](http://www.math.columbia.edu/~rf/numbertheory2.pdf)

book/src/background/polynomials.md

@@ -7,7 +7,16 @@ A(X) = a_0 + a_1 X + a_2 X^2 + a_3 X^3
 $$
 
 defines a degree-$3$ polynomial. $a_0$ is referred to as the constant term. Polynomials of
-degree $n-1$ have $n$ coefficients.
+degree $n-1$ have $n$ coefficients. We will often want to compute the result of replacing
+the formal indeterminate $X$ with some concrete value $x$, which we denote by $A(x)$.
+
+> In mathematics this is commonly referred to as "evaluating $A(X)$ at a point $x$".
+> The word "point" here stems from the geometrical usage of polynomials in the form
+> $y = A(x)$, where $(x, y)$ is the coordinate of a point in two-dimensional space.
+> However, the polynomials we deal with are almost always constrained to equal zero, and
+> $x$ will be an [element of some field](fields.md). This should not be confused
+> with points on an [elliptic curve](curves.md), which we also make use of, but never in
+> the context of polynomial evaluation.
 
 Important notes:
 
@@ -273,8 +282,8 @@ $$A(X) = \sum_{i = 0}^{n-1} A(x_i)\mathcal{L_i}(X), $$
 where $X \in \{x_0, x_1,\cdots, x_{1-n}\}.$
 
 ## References
-[^master-thm]: Dasgupta, S., Papadimitriou, C. H., & Vazirani, U. V. (2008). "Algorithms" (ch. 2). New York: McGraw-Hill Higher Education. https://people.eecs.berkeley.edu/~vazirani/algorithms/chap2.pdf
+[^master-thm]: [Dasgupta, S., Papadimitriou, C. H., & Vazirani, U. V. (2008). "Algorithms" (ch. 2). New York: McGraw-Hill Higher Education.](https://people.eecs.berkeley.edu/~vazirani/algorithms/chap2.pdf)
 
-[^ifft]: http://www.cs.ust.hk/mjg_lib/Classes/COMP3711H_Fall16/lectures/FFT_Slides.pdf
+[^ifft]: [Golin, M. (2016). "The Fast Fourier Transform and Polynomial Multiplication" [lecture notes], COMP 3711H Design and Analysis of Algorithms, Hong Kong University of Science and Technology.](http://www.cs.ust.hk/mjg_lib/Classes/COMP3711H_Fall16/lectures/FFT_Slides.pdf)
 
-[^barycentric]: Berrut, J. and Trefethen, L. (2004). "Barycentric Lagrange Interpolation." https://people.maths.ox.ac.uk/trefethen/barycentric.pdf
+[^barycentric]: [Berrut, J. and Trefethen, L. (2004). "Barycentric Lagrange Interpolation."](https://people.maths.ox.ac.uk/trefethen/barycentric.pdf)

book/src/background/upa.md

@@ -5,11 +5,11 @@ We call the field over which the circuit is defined $\mathbb{F} = \mathbb{F}_p$.
 Let $n = 2^k$, and assume that $\omega$ is a primitive root of unity of order $n$ in
 $\mathbb{F}^\times$, so that $\mathbb{F}^\times$ has a multiplicative subgroup
 $\mathcal{H} = \{1, \omega, \omega^2, \cdots, \omega^{n-1}\}$. This forms a Lagrange
-basis corresponding to the points in the subgroup.
+basis corresponding to the elements in the subgroup.
 
 ## Polynomial rules
 A polynomial rule defines a constraint that must hold between its specified columns at
-every row (i.e. at every point in the multiplicative subgroup).
+every row (i.e. at every element in the multiplicative subgroup).
 
 e.g.
 
@@ -59,7 +59,7 @@ Reference: [Generic Lookups with PLONK (DRAFT)](/LTPc5f-3S0qNF6MtwD-Tdg?view)
 
 ### Vanishing argument
 We want to check that the expressions defined by the gate constraints, permutation
-constraints and loookup constraints evaluate to zero at all points in the multiplicative
+constraints and loookup constraints evaluate to zero at all elements in the multiplicative
 subgroup. To do this, the prover collapses all the expressions into one polynomial 
 $$H(X) = \sum_{i=0}^e y^i E_i(X),$$
 where $e$ is the number of expressions and $y$ is a random challenge used to keep the