RESULTANT AND DISCRIMINANT OF POLYNOMIALS

SVANTE JANSON

Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results are well-known 19th century mathematics, but I have not investigated the history, and no references are given.

1. Resultant

Definition 1.1. Let f(x) = a_{n}x^{n} + ⋯ + a_{0} and g(x) = b_{m}x^{m} + ⋯ + b_{0} be two polynomials of degrees (at most) n and m, respectively, with coefficients in an arbitrary field F. Their resultant R(f, g) = R_{n,m}(f, g) is the element of F given by the determinant of the (m + n) × (m + n) Sylvester matrix Syl(f, g) = Syl_{n,m}(f, g) given by

(\begin{matrix}a_{n} & a_{n−1} & a_{n−2} & … & 0 & 0 & 0 \\ 0 & a_{n} & a_{n−1} & … & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & … & a_{1} & a_{0} & 0 \\ 0 & 0 & 0 & … & a_{2} & a_{1} & a_{0} \\ b_{m} & b_{m−1} & b_{m−2} & … & 0 & 0 & 0 \\ 0 & b_{m} & b_{m−1} & … & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & … & b_{1} & b_{0} & 0 \\ 0 & 0 & 0 & … & b_{2} & b_{1} & b_{0} \end{matrix})  (1.1)

where the m first rows contain the coefficients a_{n}, a_{n−1}, …, a_{0} of f shifted 0, 1, …, m − 1 steps and padded with zeros, and the n last rows contain the coefficients b_{m}, b_{m−1}, …, b_{0} of g shifted 0, 1, …, n−1 steps and padded with zeros. In other words, the entry at (i, j) equals a_{n+i−j} if 1 ≤ i ≤ m and b_{i−j} if m + 1 ≤ i ≤ m + n, with a_{i} = 0 if i > n or i < 0 and b_{i} = 0 if i > m or i < 0.

Date: September 22, 2007.

Example. If n = 3 and m = 2,

R(f, g) = |\begin{matrix}a_{3} & a_{2} & a_{1} & a_{0} & 0 \\ 0 & a_{3} & a_{2} & a_{1} & a_{0} \\ b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & b_{2} & b_{1} & b_{0} \end{matrix}| .

In the exterior algebra over F [x], we thus have

(x^{m−1}f(x)) ∧ (x^{m−2}f(x)) ∧ ⋯ ∧ f(x)

∧ (x^{n−1}g(x)) ∧ (x^{n−2}g(x)) ∧ ⋯ ∧ g(x)

= R(f, g)x^{n+m−1} ∧ x^{n+m−2} ∧ ⋯ ∧ 1, (1.2)

which can be used as an alternative form of Definition 1.1.

Remark 1.2. Typically, one assumes in Definition 1.1 that n = deg(f) and m = deg(g), i.e. that a_{n} ≠ 0 and b_{m} ≠ 0; this implies that R(f, g) is completely determined by the polynomials f and g (and it excludes the case f = 0 or g = 0). It is, however, convenient to use the slightly more general version above which also allows n and m to be regarded as given and then R(f, g) is defined for all polynomials f and g of degrees deg(f) ≤ n, deg(g) ≤ m. (See for example Remarks 1.9 and 3.4.) In this case, we may use the notation R_{n,m}(f, g) to avoid ambiguity, but usually we write just R(f, g).

Remark 1.3. It is sometimes convenient to regard a_{i} and b_{j} is indeterminates, thus regarding f and g as polynomials with coefficients in the field F (a_{n}, …, a_{0}, b_{m}, …, b_{0}). Any formula or argument that requires a_{n} ≠ 0 and b_{m} ≠ 0 then can be used; if this results in, for example, a polynomial identity involving R_{n,m}(f, g), then this formula holds also if we substitute any values in F for a_{n}, …, b_{0}.

The resultant is obviously a homogeneous polynomial of degree n + m with integer coefficients in the coefficients a_{i}, b_{j}. More precisely, we have the following. We continue to use the notations a_{i} and b_{j} for the coefficients of f and g, respectively, as in Definition 1.1.

Theorem 1.4. R_{n,m}(f, g) is a homogeneous polynomial with integer coefficients in the coefficients a_{i}, b_{j}.

(i) R_{n,m}(f, g) is homogeneous of degree m in a_{n}, …, a_{0} and degree n in b_{m}, …, b_{0}.

(ii) If a_{i} and b_{i} are regarded as having degree i, then R_{n,m}(f, g) is homogeneous of degree nm.

Proofs of this and other results in this section are given in Section 2.

Remark 1.5. If we write R_{n,m}(f, g) as a polynomial with integer coefficients for any field with characteristic 0, such as ℚ or ℂ, then the formula is valid (with the same coefficients) for every field F. (Because the coefficients are given by expanding the determinant of Syl_{n,m}(f, g) and thus have a combinatorial interpretation independent of F. Of course, for a field of characteristic p ≠ 0, the coefficients may be reduced modulo p, so they are not unique in that case.)

The main importance of the resultant lies in the following formula, which often is taken as the definition.

Theorem 1.6. Let f(x) = a_{n}x^{n}+⋯+a_{0} and g(x) = b_{m}x^{m}+⋯+b_{0} be two polynomials of degrees n and m, respectively, with coefficients in an arbitrary field F. Suppose that, in some extension of F, f has n roots ξ_{1}, …, ξ_{n} and g has m roots η_{1}, …, η_{m} (not necessarily distinct). Then

R(f, g) = a_{n}^{m}b_{m}^{n} ∏_{i=1}^{n} ∏_{j=1}^{m}(ξ_{i} − η_{j}).  (1.3)

Here and below, the roots of a polynomial are listed with multiple roots repeated according to their multiplicities. Thus every polynomial of degree n has n roots in some extension field (for example in an algebraically closed extension). Combining Theorem 1.6 and Definition 1.1, we see that the product in (1.3) lies in coefficient field F, and that it does not depend on the choice of extension field.

Theorem 1.6 implies the perhaps most important result about resultants.

Corollary 1.7. Let f and g be two non-zero polynomials with coefficients in a field F. Then f and g have a common root in some extension of F if and only if R(f, g) = 0.

It is here implicit that R = R_{n,m} with n = deg(f) and m = deg(g). Since f and g have a common root in some extension of F if and only if they have a common non-trivial (i.e., non-constant) factor in F, Corollary 1.7 can also be stated as follows.

Corollary 1.8. Let f and g be two non-zero polynomials with coefficients in a field. Then f and g have a common non-trivial factor if and only if R(f, g) = 0. Equivalently, f and g are coprime if and only if R(f, g) ≠ 0.

Remark 1.9. If n and m are fixed, we can (in the style of projective geometry) say that a polynomial f with deg(f) ≤ n has n − deg(f) roots at ∞, and similarly g has m − deg(g) roots at ∞. Thus f always has n roots and g has m, in F_{1} ∪ {∞} for some extension F_{1}. With this interpretation, Corollary 1.7 holds for all polynomials with deg(f) ≤ n and deg(g) ≤ m. (Including f = 0 and g = 0 except in the trivial case n = m = 0.)

We have also the following useful formulas related to (1.3).

Theorem 1.10. Let f(x) = a_{n}x^{n} + ⋯ + a_{0} and g(x) = b_{m}x^{m} + ⋯ + b_{0} be two polynomials with coefficients in an arbitrary field F.

(i) Suppose that f has n roots ξ_{1}, …, ξ_{n} in some extension of F. Then

R(f, g) = a_{n}^{m} ∏_{i=1}^{n} g(ξ_{i}).  (1.4)

(ii) Suppose that g has m roots η_{1}, …, η_{m} in some extension of F. Then

R(f, g) = (−1)^{nm}b_{m}^{n} ∏_{j=1}^{m} f(η_{j}).  (1.5)

In (i), f necessarily has degree n, while deg(g) ≤ m may be less than m. Similarly, in (ii), deg(g) = m and deg(f) ≤ n.

The Sylvester matrix of g and f is obtained by permuting the rows of the Sylvester matrix of f and g. The number of inversions of the permutation is nm, and it follows immediately from Definition 1.1 that

R_{m,n}(g, f) = (−1)^{nm}R_{n,m}(f, g).  (1.6)

(If deg(f) = n and deg(g) = m, (1.6) also follows from (1.3).) This kind of anti-symmetry explains why there is a factor (−1)^{nm} in (1.5) but not in (1.4).

The factorization properties in Theorem 1.10 can also be expresed as follows.

Theorem 1.11. If f_{1}, f_{2} and g are polynomials with deg(f_{1}) ≤ n_{1}, deg(f_{2}) ≤ n_{2} and deg(g) ≤ m, then

R_{n_{1}+n_{2},m}(f_{1}f_{2}, g) = R_{n_{1},m}(f_{1}, g)R_{n_{2},m}(f_{2}, g).  (1.7)

Similarly, if f, g_{1} and g_{2} are polynomials with deg(f) ≤ n , deg(g_{1}) ≤ m_{1} and deg(g_{2}) ≤ m_{2}, then

R_{n,m_{1}+m_{2}}(f, g_{1}g_{2}) = R_{n,m_{1}}(f, g_{1})R_{n,m_{2}}(f, g_{2}).  (1.8)

There is, besides (1.6), also another type of symmetry. With f(x) = a_{n}x^{n} + ⋯ + a_{0} and g(x) = b_{m}x^{m} + ⋯ + b_{0} as above, define the reversed polynomials by

f^{∗}(x) = x^{n}f(1/x) = a_{n} + a_{n−1}x + ⋯ + a_{0}x^{n},  (1.9)
g^{∗}(x) = x^{m}g(1/x) = b_{m} + b_{m−1}x + ⋯ + b_{0}x^{m}.  (1.10)

Theorem 1.12. With notations as above, for any two polynomials f and g with deg(f) ≤ n and deg(g) ≤ m,

R_{n,m}(f^{∗}, g^{∗}) = R_{m,n}(g, f) = (−1)^{nm}R_{n,m}(f, g).

As said in Remark 1.2, the standard case for the resultant is when deg(f) = n and deg(g) = m. We can always reduce to that case by the following formulas.

Theorem 1.13. (i) If deg(g) ≤ k ≤ m, then

R_{n,m}(f, g) = a_{n}^{m−k}R_{n,k}(f, g).  (1.11)

(ii) If deg(f) ≤ k ≤ n, then

R_{n,m}(f, g) = (−1)^{(n−k)m}b_{m}^{n−k}R_{k,m}(f, g).  (1.12)

Note further that if both deg(f) < n and deg(g) < m, then R_{n,m}(f, g) = 0. (Because the first column in (1.1) vanishes, or from (1.11) or (1.12).)

Theorem 1.14. Let f and g be polynomials with deg(f) ≤ n and deg(g) ≤ m. If n ≥ m and h is any polynomial with deg(h) ≤ n − m, then

R_{n,m}(f + hg, g) = R_{n,m}(f, g).  (1.13)

Similarly, if n ≤ m and h is any polynomial with deg(h) ≤ m − n, then

R_{n,m}(f, g + hf) = R_{n,m}(f, g).  (1.14)

Theorem 1.15. If f and g are polynomials of degrees n and m as above, with roots ξ_{1}, …, ξ_{n} and η_{1}, …, η_{m} in some extension field, then the resultant R(f(x), g(y − x)) (with g(y − x) regarded as a polymial in x) is a polynomial in y of degree nm with roots ξ_{i} + η_{j}, 1 ≤ i ≤ n and 1 ≤ j ≤ m. Further, R(f(x), g(y − x)) has leading coefficient a_{n}^{m}b_{m}^{n}. In particular, R(f(x), g(y − x)) is monic if both f and g are.

If deg(f) < n or deg(g) < m, but not both, then by Theorem 1.13 R(f(x), g(y − x)) is still a polynomial whose roots are given by ξ_{i} + η_{j}, where ξ_{i} runs through the roots of f and η_{j} through the roots of g (with multiplicities). If deg(f) < n and deg(g) < m, then R(f(x), g(y − x)) = 0.

Example 1.16. Let n = 1 and f(x) = ax + c. Then, if a ≠ 0, f has the single root ξ = −c/a and (1.4) yields

R_{1,m}(f, g) = a^{m}g(−c/a) = ∑_{j=0}^{m} b_{j}(−c)^{j}a^{m−j}.  (1.15)

This formula (ignoring the middle expression) holds also if a = 0 (and then simplifies to R_{1,m}(c, g) = b_{m}(−c)^{m}), for example by Remark 1.3.

Example 1.17. Let n ≥ 0 and let f(x) and g(x) be two polynomials of degree ≤ n with coefficients in a field F. Further, let a, b, c, d ∈ F.

Assume first that d ≠ 0. Then, by Theorem 1.14 and Theorem 1.4,

\begin{aligned}R_{n,n}(af + bg, cf + dg) & = R_{n,n}(af + bg − (b/d)(cf + dg), cf + dg) \\ & = R_{n,n}((a − bc/d)f, cf + dg) \\ & = (a − bc/d)^{n}R_{n,n}(f, cf + dg) \\ & = (a − bc/d)^{n}R_{n,n}(f, dg) \\ & = (ad − bc)^{n}R_{n,n}(f, g). \end{aligned} (1.16)

The final formula holds in the case d = 0 too, for example by regarding d as an indeterminate. We may write the result as

R_{n,n}(af + bg, cf + dg) = |\begin{matrix}a & b \\ c & d \end{matrix}|^{n} R_{n,n}(f, g).  (1.17)

1.1. Trivial cases. For completeness we allow n = 0 or m = 0 above. The case m = n = 0 is utterly trivial: f(x) and g(x) are constants, the Sylvester matrix (1.1) has 0 rows and columns (the empty matrix), and R_{0,0}(f, g) = 1 (by definition).

In the case m = 0, g(x) = b_{0} is constant. The Sylvester matrix is the diagonal matrix b_{0}I_{n}, where I_{n} is the n × n identity matrix, and thus R_{n,0}(f, g) = b_{0}^{n}.

Similarly, or by (1.6), if n = 0, then f(x) = a_{0} and R_{0,m}(f, g) = a_{0}^{m}. (These formulas are special cases of (1.5) and (1.4).)

Note that the formulas (1.3), (1.4), (1.5) in Theorems 1.6 and 1.10 hold also for n = 0 and m = 0, with empty products defined to be 1.

1.2. Another determinant formula.

Theorem 1.18. Let f and g be polynomials with deg(f) = n and deg(g) ≤ m. Let, for k ≥ 0, r_{k}(x) = r_{k,n−1}x^{n−1} +⋯+r_{k,0} be the remainder of x^{k}g(x) modulo f(x), i.e., x^{k}g(x) = q_{k}(x)f(x) + r_{k}(x) for some polynomial q_{k} and deg(r_{k}) ≤ n − 1. Then (where as above a_{n} is the leading coefficient of f),

R_{n,m}(f, g) = a_{n}^{m} |\begin{matrix}r_{n−1,n−1} & … & r_{n−1,0} \\ ⋮ & & ⋮ \\ r_{0,n−1} & … & r_{0,0} \end{matrix}| .  (1.18)

1.3. More on the Sylvester matrix. The following theorem extends Corollary 1.8, since it in particular says that the Sylvester matrix of f and g is singular if and only if their greatest common divisor has degree ≥ 1.

Theorem 1.19. Let f and g be two polynomials with deg(f) = n and deg(g) = m, and let h := GCD(f, g) be their greatest common divisor (i.e., a common divisor of highest degree). Then deg(h) is the corank of the Sylvester matrix Syl(f, g). In other words, the Sylvester matrix Syl(f, g) has rank n + m − deg(h).

There is also an explicit description of the left null space.

Theorem 1.20. Let f and g be two polynomials with deg(f) ≤ n and deg(g) ≤ m. Let v := (α_{m−1}, …, α_{0}, β_{n−1}, …, β_{0}) be a row vector of dimension m + n. Then v Syl(f, g) = 0 if and only if pf + qg = 0, where p(x) = α_{m−1}x^{m−1} + ⋯ + α_{0}x^{0} and q(x) = β_{n−1}x^{n−1} + ⋯ + β_{0}x^{0}.

1.4. Further examples. If n = m = 2,

R(f, g) = |\begin{matrix}a_{2} & a_{1} & a_{0} & 0 \\ 0 & a_{2} & a_{1} & a_{0} \\ b_{2} & b_{1} & b_{0} & 0 \\ 0 & b_{2} & b_{1} & b_{0} \end{matrix}| = (a_{2}b_{0} − b_{2}a_{0})^{2} − (a_{2}b_{1} − b_{2}a_{1})(a_{1}b_{0} − b_{1}a_{0}).

If n = m = 3,

R(f, g) = |\begin{matrix}a_{3} & a_{2} & a_{1} & a_{0} & 0 & 0 \\ 0 & a_{3} & a_{2} & a_{1} & a_{0} & 0 \\ 0 & 0 & a_{3} & a_{2} & a_{1} & a_{0} \\ b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & b_{3} & b_{2} & b_{1} & b_{0} \end{matrix}| .

More generally, if m = n, then

R(f, g) = |\begin{matrix}a_{n} & a_{n−1} & … & a_{0} & 0 & 0 & … & 0 \\ 0 & a_{n} & … & a_{1} & a_{0} & 0 & … & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & … & a_{n−1} & a_{n−2} & 0 & … & a_{0} \\ b_{n} & b_{n−1} & … & b_{0} & 0 & 0 & … & 0 \\ 0 & b_{n} & … & b_{1} & b_{0} & 0 & … & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & … & b_{n−1} & b_{n−2} & 0 & … & b_{0} \end{matrix}| .

2. Proofs

We begin by noting that, as said above, (1.6) holds by a permutation of the rows in (1.1).

Proof of Theorem 1.4. It is obvious that R_{n,m}(f, g) is a homogeneous polynomial with integer coefficients in the coefficients a_{i}, b_{j}, of total degree m+n. Moreover, to replace a_{i} by ta_{i} and b_{j} by ub_{j} in the Sylvester matrix means that we multiply each of the first m rows by t and each of the last n by u, and thus the determinant R_{n,m}(f, g) by t^{m}u^{n}, which shows (i). (It is here best to treat a_{i}, b_{j}, t and u as different indeterminates, and do the calculations in F (a_{0}, …, a_{n}, b_{0}, …, b_{m}, t, u).)

Similarly, (ii) follows because to replace each a_{i} by t^{i}a_{i} and each b_{j} by t^{j}b_{j} in Syl_{n,m}(f, g) yields the same result as multiplying the i:th row by t^{n+i} for i = 1, …, m and by t^{i} for i = m+1, …, m+n, and the j : th column by t^{−j}; this multiplies the determinant R_{n,m}(f, g) by t^{mn+∑_{1}^{n+m} i−∑_{1}^{n+m} j} = t^{nm}. □

Proof of Theorem 1.14. First, assume n ≥ m and deg(h) ≤ n − m. The Sylvester matrix Syl_{n,m}(f + hg, g) is obtained from Syl_{n,m}(f, g) by row operations that do not change its determinant R_{n,m}. (If h(x) = c_{l}x^{l} + ⋯ + c_{0}, add c_{k} times row n + i − k to row i, for i = 1, …, m and k = 0, …, l.)

The second part follows similarly, or by the first part and (1.6). □

Proof of Theorem 1.13. (i). Suppose that deg(g) < m, so b_{m} = 0. Then the first column of the Sylvester matrix (1.1) is 0 except for its first element a_{n}, and the submatrix of Syl_{n,m}(f, g) obtained by deleting the first row and column equals Syl_{n,m−1}(f, g). Hence, by expanding the determinant R_{n,m}(f, g) along the first column,

R_{n,m}(f, g) = a_{n}R_{n,m−1}(f, g).

The formula (1.11) now follows for k = m, m − 1, …, 0 by backwards induction.

(ii). By (1.6) and part (i),

\begin{aligned}R_{n,m}(f, g) & = (−1)^{nm}R_{m,n}(g, f) = (−1)^{nm}b_{m}^{n−k}R_{m,k}(g, f) \\ & = (−1)^{nm−km}b_{m}^{n−k}R_{k,m}(f, g). \end{aligned}□

Proof of Theorems 1.6 and 1.10. We prove these theorems together by induction on n + m.

Assume that deg(f) = n and deg(g) = m. Then, at least in some extension field, f(x) = a_{n} ∏_{i=1}^{n}(x − ξ_{i}) and g(x) = b_{m} ∏_{j=1}^{m}(x − η_{j}), and (1.3) is equivalent to both (1.4) and (1.5). Assume by induction that these formulas hold for all smaller values of n + m (and all polynomials of these degrees).

Case 1. First, suppose 0 < n = deg(f) ≤ m = deg(g). Divide g by f to obtain polynomials q and r with g = qf + r and deg(r) < deg(f) = n. Note that

deg(q) = deg(qf) − deg(f) = deg(g − r) − n = m − n.

By Theorem 1.14,

R_{n,m}(f, g) = R_{n,m}(f, g − qf) = R_{n,m}(f, r).  (2.1)

Case 1a. Suppose that r ≠ 0 and let k := deg(r) ≥ 0. By Theorem 1.13 and the inductive hypothesis in the form (1.4),

R_{n,m}(f, r) = a_{n}^{m−k}R_{n,k}(f, r) = a_{n}^{m−k}a_{n}^{k} ∏_{i=1}^{n} r(ξ_{i}) = a_{n}^{m} ∏_{i=1}^{n} g(ξ_{i}),

since g(ξ_{i}) = q(ξ_{i})f(ξ_{i}) + r(ξ_{i}) = r(ξ_{i}), which verifies (1.4) and thus (1.3).

Case 1b. Suppose now that r = 0, so g = qf, but n > 0. Then Syl_{n,m}(f, r) = Syl_{n,m}(f, 0) has the last n rows identically 0, so R_{n,m}(f, r) = 0, and R_{n,m}(f, g) = 0 by (2.1). Further, g(ξ_{1}) = q(ξ_{1})f(ξ_{1}) = 0 so ξ_{1} is a root of g too, and the right hand side of (1.3) vanishes too. Hence, (1.3) holds.

Case 2. Suppose that n = 0. As remarked in Subsection 1.1, R_{0,m}(f, g) = a_{0}^{m}, which agrees with (1.3). (This includes the case n = m = 0 that starts the induction.)

Case 3. Suppose that m = deg(g) < n = deg(f). This is reduced to Case 1 or 2 by (1.6).

This completes the induction, and the proof of Theorem 1.6. It remains to verify Theorem 1.10(i),(ii) also in the cases deg(g) < m and deg(f) < n, respectively. This follows by Theorem 1.13, as in the proof of Case 1a above, or by Remark 1.3. □

Proof of Corollaries 1.7 and 1.8. Immediate from (1.3), using the fact on common factors stated before Corollary 1.8. □

Proof of Theorem 1.11. By the argument in Remark 1.3, we may assume that deg(g) = m, so g has m roots in some extension of F, and then (1.7) follows from (1.5). Similarly, (1.8) follows from (1.4). □

Proof of Theorem 1.12. The Sylvester matrix Syl_{n,m}(f^{∗}, g^{∗}) is obtained from Syl_{m,n}(g, f) by reversing the order of both rows and columns, and thus they have the same determinant. □

Proof of Theorem 1.15. We have g(x) = b_{m} ∏_{j=1}^{m}(x − η_{j}) and thus

g(y − x) = b_{m} ∏_{j=1}^{m}(y − x − η_{j}) = (−1)^{m}b_{m} ∏_{j=1}^{m}(x − y + η_{j}).

Thus, by (1.4) (or Theorem 1.6),

\begin{aligned}R(f(x), g(y − x)) & = a_{n}^{m}(−1)^{nm}b_{m}^{n} ∏_{i=1}^{n} ∏_{j=1}^{m}(ξ_{i} − y + η_{j}) \\ & = a_{n}^{m}b_{m}^{n} ∏_{i=1}^{n} ∏_{j=1}^{m}(y − ξ_{i} − η_{j}). \end{aligned}  □

Proof of Theorem 1.18. We work in the exterior algebra over F (x), using (1.2). Let D be the determinant in (1.18); thus

r_{n−1}(x) ∧ ⋯ ∧ r_{0}(x) = Dx^{n−1} ∧ ⋯ ∧ x^{0}.  (2.2)

For k ≤ n − 1, x^{k}g(x) − r_{k}(x) = q_{k}(x)f(x) has degree ≤ n + m − 1 and is thus a linear combination of f(x), xf(x), …, x^{m−1}f(x); hence, using (2.2) and (1.2) (with g(x) replaced by 1),

(x^{m−1}f(x)) ∧ ⋯ ∧ f(x) ∧ (x^{n−1}g(x)) ∧ ⋯ ∧ g(x)
= (x^{m−1}f(x)) ∧ ⋯ ∧ f(x) ∧ r_{n−1}(x) ∧ ⋯ ∧ r_{0}(x)
= (x^{m−1}f(x)) ∧ ⋯ ∧ f(x) ∧ Dx^{n−1} ∧ ⋯ ∧ x^{0}
= DR_{n,m}(f, 1)x^{n+m−1} ∧ ⋯ ∧ 1.

Consequently, using (1.2) again,

R_{n,m}(f, g) = DR_{n,m}(f, 1).

Finally, by Theorem 1.13, or by (1.4),

R_{n,m}(f, 1) = a_{n}^{m}R_{n,0}(f, 1) = a_{n}^{m}.  □

Proof of Theorem 1.19. Note first that Syl_{n,m}(f, g) and Syl_{m,n}(g, f) have the same rank and corank, so we may interchange f and g. We may thus assume n ≥ m.

In this case, we may as in the proof of Theorem 1.14 for any polynomial q with deg(q) ≤ n − m obtain Syl_{n,m}(f − qg, g) from Syl_{n,m}(f, g) by row operations that do not change the rank and corank. In particular, we may replace f by the remainder r obtained when dividing f by g. Then deg(r) < m ≤ n and the first column of Syl_{n,m}(r, g) has a single non-zero element, b_{m} in row m + 1. We may thus delete the first column and the m + 1:th row without changing the corank, and this yields Syl_{n−1,m}(r, g). (Cf. the proof of Theorem 1.13.) Repeating, we see that if r ≠ 0 and k = deg(r), then Syl_{n,m}(f, g) has the same corank as Syl_{k,m}(r, g) and Syl_{m,k}(g, r). We repeat from the start, by dividing g by r and so on; this yields the Euclidean algorithm for finding the GCD h, and we finally end up with the Sylvester matrix Syl_{k,l}(0, h), for some k ≥ 0 and l = deg(h), which evidently has corank l since the first l rows are 0 and the last k are independent, as is witnessed by the lower left k × k minor which is triangular.

(Alternatively, Theorem 1.19 follows easily from Theorem 1.20.) □

Proof of Theorem 1.20. Let v Syl_{n,m}(f, g) = (γ_{1}, …, γ_{n+m}). Then, for j = 1, …, m + n, with a_{k} and b_{k} defined for all integers k as at the end of Definition 1.1,

γ_{j} = ∑_{i=1}^{m} α_{m−i}a_{n+i−j} + ∑_{i=m+1}^{m+n} β_{m+n−i}b_{i−j},

which equals the coefficient of x^{m+n−j} in pf + qg. □

3. Discriminant

Several different normalizations of the discriminant of a polynomial are used by different authors, differing in sign and in factors that are powers of the leading coefficient of the polynomial. One natural choice is the following.

Definition 3.1. Let f be a polynomial of degree n ≥ 1 with coefficients in an arbitrary field F. Let F_{1} be an extension of F where f splits, and let ξ_{1}, …, ξ_{n} be the roots of f in F_{1} (taken with multiplicities). Then the (normalized) discriminant of f is

Δ_{0}(f) := ∏_{1≤i<j≤n}(ξ_{i} − ξ_{j})^{2}.  (3.1)

Note that such a field F_{1} always exists, for example an algebraic closure of F will do, and that, e.g. by Theorem 3.3 below, Δ_{0}(f) ∈ F and does not depend on the choice of F_{1}. (This also follows from the fact that Δ_{0}(f) is a symmetric polynomial in ξ_{1}, …, ξ_{n}, and thus by a well-known fact a polynomial in the elementary symmetric polynomials σ_{k}(ξ_{1}, …, ξ_{n}) = (−1)^{k}a_{n−k}/a_{n}, k = 1, …, n.)

Note further that Δ_{0}(cf) = Δ_{0}(f) for any constant c ≠ 0.

However, while the definition of Δ_{0} is simple and natural, Δ_{0} is particularly useful for monic polynomials. In general, it is often more convenient to use the following version, which by Theorem 3.5 below is a polynomial in the coefficients of f. (This is the most common version of the discriminant. The names ’normalized’ and ’standard’ are my own.)

Definition 3.2. Let f = a_{n}x^{n} + ⋯ + a_{0} be a polynomial of degree n ≥ 1 with coefficients in an arbitrary field F. Then the (standard) discriminant of f is

Δ(f) := a_{n}^{2n−2}Δ_{0}(f) = a_{n}^{2n−2} ∏_{1≤i<j≤n}(ξ_{i} − ξ_{j})^{2},  (3.2)

where as above ξ_{1}, …, ξ_{n} are the roots of f in some extension F_{1} of F.

The discriminant can also be defined as the resultant of f and its derivative f′, with a suitable normalizing factor, as is stated more precisely in the following theorem.

Theorem 3.3. Let f = a_{n}x^{n} +⋯+a_{0} be a polynomial of degree n ≥ 1 with coefficients in an arbitrary field F. Then the discriminant of f is given by

Δ(f) = (−1)^{n(n−1)/2}a_{n}^{−1}R(f, f′)  (3.3)

and thus

Δ_{0}(f) = (−1)^{n(n−1)/2}a_{n}^{−(2n−1)}R(f, f′).  (3.4)

Remark 3.4. To be precise, we should write R_{n,n−1}(f, f′) in this theorem. Typically, deg(f′) = deg(f) − 1 = n − 1 and we may then write R(f, f′) without any ambiguity. (For example, always when F has characteristic 0, such as ℝ and ℂ.) However, if F has characteristic p > 0 and p|n, then deg(f′) < n − 1. In this case, if deg(f′) = k, then by Theorem 1.13, R_{n,n−1}(f, f′) = a_{n}^{n−1−k}R_{n,k}(f, f′) and thus

Δ(f) = (−1)^{n(n−1)/2}a_{n}^{n−k−2}R_{n,k}(f, f′),  (3.5)
Δ_{0}(f) = (−1)^{n(n−1)/2}a_{n}^{−n−k}R_{n,k}(f, f′).  (3.6)

Proof of Theorem 3.3. Let f have roots ξ_{1}, …, ξ_{n} (in some extension field). Then f(x) = a_{n} ∏_{i=1}^{n}(x − ξ), and thus f′(ξ_{i}) = a_{n} ∏_{j≠i}(ξ_{i} − ξ_{j}). Consequently, by (1.4),

\begin{aligned}R_{n,n−1}(f, f′) & = a_{n}^{n−1+n} ∏_{i=1}^{n} ∏_{j≠i}(ξ_{i} − ξ_{j}) = a_{n}^{2n−1} ∏_{1≤i<j≤n}(ξ_{i} − ξ_{j})(ξ_{j} − ξ_{i}) \\ & = (−1)^{n(n−1)/2}a_{n}^{2n−1}Δ_{0}(f) = (−1)^{n(n−1)/2}a_{n}Δ(f). \end{aligned}□

Theorem 3.5. Δ(f) is a homogeneous polynomial with integer coefficients in the coefficients a_{0}, …, a_{n} of f. Further, with n = deg(f),

(i) Δ(f) is homogeneous of degree 2n − 2 in a_{0}, …, a_{n}.

(ii) If a_{i} is regarded as having degree i, then Δ(f) is homogeneous of degree n(n − 1).

Proof. The derivative f′(x) = b_{n−1}x^{n−1} + ⋯ + b_{0} with b_{j} = (j + 1)a_{j+1}. Hence all entries of the Sylvester matrix Syl_{n,n−1}(f, f′) are integer multiples of a_{0}, …, a_{n}, and thus R(f, f′) is a homogeneous polynomial with integer coefficients in a_{0}, …, a_{n}. Moreover, the only (possibly) non-zero entries in the first column of Syl_{n,n−1}(f, f′) are a_{n} and b_{n−1} = na_{n}; hence R(f, f′) is a multiple of a_{n}, and Δ(f, f′) is also such a polynomial by (3.3).

Since R(f, f′) has total degree n + n − 1, this also shows that Δ(f) has degree 2n−2. Alternatively, this follows from Definition 3.2, since replacing a_{i} by ta_{i} for all i does not change the roots ξ_{1}, …, ξ_{n} of f.

For (ii), note that if f_{t}(x) = ∑_{i=0}^{n} a_{i}t^{i}x^{i}, for an indeterminate t, then f_{t} has roots t^{−1}ξ_{1}, …, t^{−1}ξ_{n}, and Definition 3.2 yields

Δ(f_{t}) = (t^{n}a_{n})^{2n−2}t^{−n(n−1)} ∏_{1≤i<j≤n}(ξ_{i} − ξ_{j})^{2} = t^{n(n−1)}Δ(f).

(Alternatively, (ii) is easily derived from (3.3) and Theorem 1.4(i),(ii).) □

Remark 3.6. As for the resultant, see Remark 1.5, the integer coefficients of Δ(f) do not depend on the field F (except for the obvious non-uniqueness when char(F) ≠ 0).

The perhaps most important use of the discriminant is the following immediate consequence of Definitions 3.1 and 3.2.

Theorem 3.7. Let f be a polynomial of degree n ≥ 1 with coefficients in a field F. Then

Δ_{0}(f) = 0 ⟺ Δ(f) = 0 ⟺ f has a double root in some extension of F .

Equivalently, f has n distinct roots in some extension field if and only if the discriminant is ≠ 0.

By Theorem 3.5, Δ(f) for f of a given degree n ≥ 1 is a polynomial in a_{0}, …, a_{n}; we can apply this polynomial also when a_{n} = 0, i.e., to polynomials f of degree < n. To avoid confusion, we denote this polynomial in the coefficients a_{0}, …, a_{n} by Δ^{(n)}(f), defined for all polynomials f = a_{n}x^{n}+⋯+a_{0} of degree ≤ n. Thus Δ^{(n)}(f) = Δ(f) when a_{n} ≠ 0. This polynomial has a simple symmetry.

Theorem 3.8. If f = a_{n}x^{n} + ⋯ + a_{0} is a polynomial of degree ≤ n and f^{∗} is defined by (1.9), then

Δ^{(n)}(f^{∗}) = Δ^{(n)}(f).  (3.7)

In particular, if f has degree n and a_{0} ≠ 0, then

Δ(f^{∗}) = Δ(f).  (3.8)

Proof. Suppose first that a_{n} ≠ 0 and a_{0} ≠ 0. Let f have roots ξ_{1}, …, ξ_{n} in some extension field; these roots are non-zero and f^{∗} has the roots ξ_{1}^{−1}, …, ξ_{n}^{−1} and leading coefficient a_{0} = a_{n}ξ_{1} ⋯ ξ_{n}. Hence, by Definition 3.2,

Δ(f^{∗}) = a_{0}^{2n−2} ∏_{1≤i<j≤n}(ξ_{i}^{−1} − ξ_{j}^{−1})^{2} = a_{n}^{2n−2} ∏_{1≤i<j≤n}(ξ_{j} − ξ_{i})^{2} = Δ(f),

which proves (3.8). In particular, this holds if we regard a_{0}, …, a_{n} as indeterminates, and thus (3.7) follows in general because both sides are polynomials in a_{0}, …, a_{n}. □

We give another simple consequence of the definition.

Theorem 3.9. If f and g are polynomials of degrees n and m ≥ 1, then

Δ(fg) = Δ(f)Δ(g)R(f, g)^{2}.  (3.9)

Proof. This follows from Definition 3.2 and Theorem 1.6.

Alternatively, by Theorems 1.11, 1.14 and 1.11 again,

\begin{aligned}R(fg, (fg)′) & = R(fg, f′g + fg′) = R(f, f′g + fg′)R(g, f′g + fg′) \\ & = R(f, f′g)R(g, fg′) = R(f, f′)R(f, g)R(g, f)R(g, g′), \end{aligned}

and the result follows by (3.3) and (1.6). □

As said above, Δ^{(n)}(f) = Δ(f) when a_{n} ≠ 0. In the opposite case a_{n} = 0, we have the following simple formula, which can be regarded as a relation between discriminants for polynomials of different degrees. (See the examples in (4.1) and (4.3), or, more complicated, in Examples 4.7 and 4.3.)

Theorem 3.10. If a_{n} = 0, then

Δ^{(n)}(f) = a_{n−1}^{2}Δ^{(n−1)}(f).  (3.10)

In particular, if a_{n} = a_{n−1} = 0, then Δ^{(n)}(f) = 0.

Proof. Assume first a_{n−1} ≠ 0 and a_{0} ≠ 0. Let g(x) = a_{n−1}x^{n−1} + ⋯ + a_{0} (this is the same as f(x), but we regard it as a polynomial of degree n − 1), and define f^{∗} by (1.9) and g^{∗} by (1.10), with m replaced by n − 1. Then f^{∗}(x) = xg^{∗}(x), where f^{∗} has degree n and g^{∗}(x) degree n − 1. Trivially, Δ(x) = 1, and Example 1.16 shows R_{1,n−1}(x, g^{∗}) = g^{∗}(0) = a_{n−1}. Hence, Theorems 3.8 and 3.9 yield

Δ^{(n)}(f) = Δ(f^{∗}) = Δ(xg^{∗}) = Δ(g^{∗})a_{n−1}^{2} = a_{n−1}^{2}Δ^{(n−1)}(g),

which shows (3.10) in the case a_{n−1}, a_{0} ≠ 0. The general case follows, because both sides of (3.10) are polynomials in a_{0}, …, a_{n−1}. (An alternative proof without using inversion and Theorem 3.8 is givven in Appendix A.) □

Remark 3.11. If we fix n ≥ 1 and as in Remark 1.9 say that a polynomial f with deg(f) ≤ n has n − deg(f) roots at ∞, then Theorems 3.7 and 3.10 show that Δ^{(n)}(f) = 0 if and only if f has a double root (or more precisely, a multiple root) in F_{1} ∪ {∞} for some extension F_{1} (and in any extension where f splits).

Remark 3.12. If f = a_{n}x^{n} + ⋯ + a_{0} is a polynomial of degree n with non-zero a_{0}, …, a_{n−1}, define

Δ^{∗}(f) := ∏_{i=0}^{n−1} a_{i}^{−2} · Δ(f).  (3.11)

Theorem 3.10 shows that if we, more generally, for f of degree ≤ n define

Δ^{(n)∗}(f) := ∏_{i=0}^{n−1} a_{i}^{−2} · Δ^{(n)}(f),  (3.12)

then, whenever deg(f) < n,

Δ^{(n)∗}(f) = Δ^{(n−1)∗}(f).  (3.13)

It is here best to regard the coefficients a_{i} as indeterminates; then Δ^{(n)∗}(f) is a Laurent polynomial in a_{0}, …, a_{n}, and (3.13) shows that there is a single Laurent series Δ^{∗} in the infinitely many indeterminates a_{0}, a_{1}, … such that if f is a polynomial of any degree n ≥ 1, then Δ^{∗}(f) is obtained from this series Δ^{∗} by substituting a_{i} = 0 for i > n. (This has to be done with some care since also negative powers appear, but each term containing a negative power a_{i}^{−α_{i}} with i > n contains also a positive power a_{j}^{α_{j}} with j > i > n, so there is no real problem; we simply delete all terms containing some non-zero power of some a_{i} with i > n.) We may regard Δ^{∗} as a universal discriminant. (Or as a mere curiosity.)

It follows from Theorem 3.5 that the monomials that appear in Δ^{∗} have integer coefficients and all have the form ∏_{i=0}^{k} a_{i}^{α_{i}} with ∑_{i} α_{i} = −2 and ∑_{i} iα_{i} = 0; except for the term a_{0}^{−2}, they further have α_{k} > 0 if k is chosen minimal. See Example 4.11.

4. Examples of discriminants

Example 4.1 (n = 1). If f = ax + b, then trivially Δ(f) = Δ(f_{0}) = 1.

Example 4.2 (n = 2). If f(x) = ax^{2} + bx + c, then Theorem 3.3 yields

Δ(f) = −a^{−1}R(f, f′) = −a^{−1} |\begin{matrix}a & b & c \\ 2a & b & 0 \\ 0 & 2a & b \end{matrix}| = b^{2} − 4ac  (4.1)

and

Δ_{0}(f) = a^{−2}Δ(f) = \frac{b^{2} − 4ac}{a^{2}} = (\frac{b}{a})^{2} − 4\frac{c}{a}.  (4.2)

Note that the standard formula for finding the roots of ax^{2} + bx + c = 0 can be written

x_{±} = \frac{b}{2a} ± \frac{1}{2}\sqrt{Δ_{0}(f)} = \frac{b ± \sqrt{Δ(f)}}{2a}.

Example 4.3 (n = 3). If f(x) = ax^{3} + bx^{2} + cx + d, then Theorem 3.3 yields

\begin{aligned}Δ(f) & = −a^{−1}R(f, f′) = −a^{−1} |\begin{matrix}a & b & c & d & 0 \\ 0 & a & b & c & d \\ 3a & 2b & c & 0 & 0 \\ 0 & 3a & 2b & c & 0 \\ 0 & 0 & 3a & 2b & c \end{matrix}| \\ & = b^{2}c^{2} − 4ac^{3} − 4b^{3}d + 18abcd − 27a^{2}d^{2}. \end{aligned}  (4.3)

Example 4.4 (n = 3). If f(x) = x^{3} + bx^{2} + cx + d is monic, (4.3) simplifies to

Δ_{0}(f) = Δ(f) = −R(f, f′) = b^{2}c^{2} − 4c^{3} − 4b^{3}d + 18bcd − 27d^{2}.

Example 4.5 (n = 3). For f(x) = x^{3} +px+q, without second degree term, (4.3) simplifies further to

Δ_{0}(f) = Δ(f) = −4p^{3} − 27q^{2}.

Example 4.6 (n = 3). The polynomial f(x) = 4x^{3} − g_{2}x − g_{3} is important in the theory of the Weierstrass elliptic functions. Its discriminant is, by (4.3),

Δ(4x^{3} − g_{2}x − g_{3}) = 16g_{2}^{3} − 432g_{3}^{2}.

Equivalently, Δ_{0}(4x^{3} − g_{2}x − g_{3}) = \frac{1}{16}g_{2}^{3} − \frac{27}{16}g_{3}^{2}. In this context, it is customary to change the normalization and define the discriminant as

16Δ_{0}(f) = \frac{1}{16}Δ(f) = g_{2}^{3} − 27g_{3}^{2}.

Example 4.7 (n = 4). If f(x) = ax^{4} +bx^{3} +cx^{2} +dx+e, then Theorem 3.3 yields

\begin{aligned}Δ(f) & = a^{−1}R(f, f′) = a^{−1} |\begin{matrix}a & b & c & d & e & 0 & 0 \\ 0 & a & b & c & d & e & 0 \\ 0 & 0 & a & b & c & d & e \\ 4a & 3b & 2c & d & 0 & 0 & 0 \\ 0 & 4a & 3b & 2c & d & 0 & 0 \\ 0 & 0 & 4a & 3b & 2c & d & 0 \\ 0 & 0 & 0 & 4a & 3b & 2c & d \end{matrix}| \\ & = b^{2}c^{2}d^{2} − 4 b^{2}c^{3}e − 4 b^{3}d^{3} + 18 b^{3}cde − 27 b^{4}e^{2} − 4 ac^{3}d^{2} + 16 ac^{4}e \\ & + 18 abcd^{3} − 80 abc^{2}de − 6 ab^{2}d^{2}e + 144 ab^{2}ce^{2} − 27 a^{2}d^{4} \\ & + 144 a^{2}cd^{2}e − 128 a^{2}c^{2}e^{2} − 192 a^{2}bde^{2} + 256 a^{3}e^{3}. \end{aligned} ; (4.4)

Example 4.8 (n = 4). If f(x) = x^{4} + bx^{3} + cx^{2} + dx + e is monic, then (4.4) simplifies slightly to

Δ_{0}(f) = Δ(f) = R(f, f′)
= b^{2}c^{2}d^{2} − 4 b^{2}c^{3}e − 4 b^{3}d^{3} + 18 b^{3}cde − 27 b^{4}e^{2} − 4 c^{3}d^{2}
+ 16 c^{4}e + 18 bcd^{3} − 80 bc^{2}de − 6 b^{2}d^{2}e + 144 b^{2}ce^{2}
− 27 d^{4} + 144 cd^{2}e − 128 c^{2}e^{2} − 192 bde^{2} + 256 e^{3}.

Example 4.9 (n = 4). If f(x) = x^{4} + px^{2} + qx + r is monic and without third degree term, then (4.4) simplifies further to

Δ_{0}(f) = Δ(f) = −4 p^{3}q^{2} − 27 q^{4} + 16 p^{4}r + 144 pq^{2}r − 128 p^{2}r^{2} + 256 r^{3}.

Example 4.10. Let f(x) = x^{n} + px + q for some n ≥ 2. Then f′(x) = nx^{n−1} + p and, using Theorem 1.14 with h(x) = −x/n, Theorem 1.13 and Example 1.16, at least if F has characteristic 0,

(−1)^{n(n−1)/2}Δ(f) = R_{n,n−1}(f, f′) = R_{n,n−1}(x^{n} + px + q, nx^{n−1} + p)
= R_{n,n−1}(p(1 − 1/n)x + q, nx^{n−1} + p)

= (−1)^{(n−1)^{2}}n^{n−1}R_{1,n−1}(p(1 − 1/n)x + q, nx^{n−1} + p)

= (−1)^{(n−1)}n^{n−1}(n(−q)^{n−1} + p(p(1 − 1/n))^{n−1})
= n^{n}q^{n−1} + (−1)^{n−1}(n − 1)^{n−1}p^{n}.

Since the right hand side is a polynomial in p and q with integer coefficient, the final formula holds for all fields and all n ≥ 2. Consequently,

Δ_{0}(f) = Δ(f) = (−1)^{(n−1)(n−2)/2}(n − 1)^{n−1}p^{n} + (−1)^{n(n−1)/2}n^{n}q^{n−1}.

Note the special cases in Examples 4.2, 4.5 and 4.9 (with p = 0). The next case is the quintic in Bring’s form:

Δ_{0}(x^{5} + px + q) = Δ(x^{5} + px + q) = 4^{4}p^{5} + 5^{5}q^{4}.

Example 4.11. It follows from Remark 3.12 and Example 4.7 that

\begin{aligned}Δ^{∗} = a_{0}^{−2} & − 4 a_{2}a_{1}^{−2}a_{0}^{−1} − 4 a_{3}a_{2}^{−2}a_{1}a_{0}^{−2} + 18 a_{3}a_{2}^{−1}a_{1}^{−1}a_{0}^{−1} − 27 a_{3}^{2}a_{2}^{−2}a_{1}^{−2} \\ & − 4 a_{4}a_{3}^{−2}a_{2}a_{0}^{−2} + 16 a_{4}a_{3}^{−2}a_{2}^{2}a_{1}^{−2}a_{0}^{−1} + 18 a_{4}a_{3}^{−1}a_{2}^{−1}a_{1}a_{0}^{−2} \\ & − 80 a_{4}a_{3}^{−1}a_{1}^{−1}a_{0}^{−1} − 6 a_{4}a_{2}^{−2}a_{0}^{−1} + 144 a_{4}a_{2}^{−1}a_{1}^{−2} \\ & − 27 a_{4}^{2}a_{3}^{−2}a_{2}^{−2}a_{1}^{2}a_{0}^{−2} + 144 a_{4}^{2}a_{3}^{−2}a_{2}^{−1}a_{0}^{−1} − 128 a_{4}^{2}a_{3}^{−2}a_{1}^{−2} \\ & − 192 a_{4}^{2}a_{3}^{−1}a_{2}^{−2}a_{1}^{−1} + 256 a_{4}^{3}a_{3}^{−2}a_{2}^{−2}a_{1}^{−2}a_{0} + …, \end{aligned} ; (4.5)

where the omitted terms have at least one factor a_{k} with k ≥ 5. In particular, if f(x) = a_{4}x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0} has degree at most 4, this formula without “… ” is an exact formula for Δ^{∗}(f). Setting a_{4} = 0 we find that if deg(f) ≤ 3, then

Δ^{∗}(f) = a_{0}^{−2} − 4 a_{2}a_{1}^{−2}a_{0}^{−1} − 4 a_{3}a_{2}^{−2}a_{1}a_{0}^{−2} + 18 a_{3}a_{2}^{−1}a_{1}^{−1}a_{0}^{−1} − 27 a_{3}^{2}a_{2}^{−2}a_{1}^{−2}, which is equivalent to (4.3). Similarly, setting also a_{3} = 0, if deg(f) ≤ 2, then

Δ^{∗}(f) = a_{0}^{−2} − 4 a_{2}a_{1}^{−2}a_{0}^{−1},

which is equivalent to (4.1).

5. Discriminants for real polynomials

If f is real and of degree n, then its n roots in ℂ consist of n−2ν real roots and ν pairs ξ_{i}, \overline{ξ_{i}} of complex (non-real) roots, for some ν with 0 ≤ ν ≤ n/2.

Theorem 5.1. If f is a real polynomial of degree n ≥ 1 with n − 2ν real roots and ν pairs of complex (non-real) roots, and all roots are distinct and thus Δ(f) ≠ 0, then

sign(Δ(f)) = sign(Δ_{0}(f)) = (−1)^{ν}.  (5.1)

Proof. This is easily seen directly from Definitions 3.1 and 3.2, by suitably pairing terms.

Alternatively, we may factor f into its irreducible real factors f_{1}, …, f_{n−2ν}, g_{1}, …, g_{ν}, where deg(f_{i}) = 1 and deg(g_{j}) = 2, and note that Theorem 3.9 and induction shows

sign(Δ(f)) = ∏_{i=1}^{n−2ν} sign(Δ(f_{i})) ∏_{j=1}^{ν} sign(Δ(g_{j})).

Further, each Δ(f_{i}) = 1, while Δ(g_{j}) < 0 by Definition 3.2, since g_{j} has two roots ξ and \overline{ξ} with (ξ − \overline{ξ})^{2} < 0. □

For example, this leads to the following classifications for low degrees. (In these examples, “complex” means “non-real”.)

Example 5.2 (n = 2). For a real quadratic polynomial f, • Δ(f) > 0 ⟺ Δ_{0}(f) > 0 ⟺ f has two distinct real roots;

• Δ(f) < 0 ⟺ Δ_{0}(f) < 0 ⟺ f has no real root and two conjugate complex roots.

• Δ(f) = 0 ⟺ Δ_{0}(f) = 0 ⟺ f has a double real root;

Example 5.3 (n = 3). For a real cubic polynomial f, • Δ(f) > 0 ⟺ Δ_{0}(f) > 0 ⟺ f has 3 distinct real roots;

• Δ(f) < 0 ⟺ Δ_{0}(f) < 0 ⟺ f has 1 real root and 2 conjugate complex roots.

• Δ(f) = 0 ⟺ Δ_{0}(f) = 0 ⟺ f has either a triple real root, or one double real root and one single real root;

Example 5.4 (n = 4). For a real quartic polynomial f,

• Δ(f) > 0 ⟺ Δ_{0}(f) > 0 ⟺ f has either 4 distinct real roots, or 4 complex roots (in two conjugate pairs);

• Δ(f) < 0 ⟺ Δ_{0}(f) < 0 ⟺ f has 2 real roots and 2 conjugate complex roots.

• Δ(f) = 0 ⟺ Δ_{0}(f) = 0 ⟺ f has 1 quadruple real root, or 2 real roots, one triple and one single, or 2 double real roots, or 3 real roots, one double and two single, or 1 double real root and 2 conjugate complex roots, or 2 conjugate complex double roots.

Appendix A

Alternative proof of Theorem 3.10. Assume first a_{n−1} ≠ 0; thus f(x) = a_{n−1}x^{n−1} + ⋯ + a_{0} has degree n − 1.

Let f_{ε}(x) = (−εx+1)f(x) for an indeterminate ε. Trivially, Δ(−εx+1) = 1, and Example 1.16 shows

R(−εx + 1, f) = (−ε)^{n−1}f(1/ε) = (−1)^{n−1}f^{∗}(ε)

with f^{∗} defined by (1.9) with n replaced by n−1. Hence, Theorem 3.9 yields

Δ^{(n)}(f_{ε}) = Δ(f_{ε}) = Δ(f)(f^{∗}(ε))^{2} = Δ^{(n−1)}(f)f^{∗}(ε)^{2}.

Both sides are polynomials in a_{0}, …, a_{n−1} and ε, so we may here put ε = 0 and obtain Δ^{(n)}(f) = Δ^{(n−1)}(f)f^{∗}(0)^{2}. Since f^{∗}(0) = a_{n−1}, this proves the result when a_{n−1} ≠ 0. The general case follows, because both sides of (3.10) are polynomials in a_{0}, …, a_{n−1}. □

Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden

   E-mail address: svante.janson@math.uu.se
   URL: http://www.math.uu.se/∼svante/