Q. Discriminant of hyperelliptic curve \(y^2 = f(x)\).

Q: What is the discriminant of the hyperelliptic curve y^2 = f(x)?

The discriminant of the curve is (up to conventional normalization) the polynomial discriminant of f. It vanishes exactly when f has a repeated root, so it detects affine singularities of the model y^2=f(x).

Q: How is the polynomial discriminant of f expressed in terms of its roots?

If \deg f = n and f(x) = a_n \prod_{i=1}^n (x-r_i), then \operatorname{Disc}(f) = a_n^{2n-2} \prod_{i<j} (r_i-r_j)^2. It is zero precisely when some r_i = r_j.

Q: How can I compute the discriminant using resultants?

Use the resultant of f and f': \operatorname{Disc}(f) = \frac{\operatorname{Res}(f,f')}{a_n}. Computer algebra systems (Sage, Magma, PARI/GP, sympy) compute Res and Disc directly.

Q: When is the hyperelliptic curve smooth?

The affine model y^2=f(x) is smooth iff \operatorname{Disc}(f)\neq 0. repeated root r gives the singular point (r,0). (One must also check the point(s) at infinity for completeness in projective models.)

Q: How does the discriminant change under the linear change x\mapsto ax+b?

Under x\mapsto ax+b the discriminant scales by a^{n(n-1)} where n=\deg f. Translation by b does not change the value other than this scaling effect from a.

Q: What happens to the discriminant when you twist by a scalar c: y^2 = c\cdot f(x)?

What happens to the discriminant when you twist by a scalar c: y^2 = c\cdot f(x)?

Q: How does the degree of f determine the genus?

If \deg f=n and f is separable, the genus is g=\lfloor (n-1)/2\rfloor. Typical models use n=2g+1 or n=2g+2.

Q: What does vanishing of the discriminant modulo a prime p mean for reduction?

If \operatorname{Disc}(f)\equiv 0\pmod p the reduction mod p of the given model is singular: the curve has bad reduction at p. The valuation v_p(\operatorname{Disc}(f)) helps measure severity; a minimal discriminant refines this.

Answer

Let \(f(x)=a_n \prod_{i=1}^n (x-r_i)\). The discriminant of the polynomial \(f\) is
\[
\mathrm{Disc}(f)=a_n^{2n-2}\prod_{1\le i<j\le n}(r_i-r_j)^2.
\]
For the hyperelliptic curve \(y^2=f(x)\) one takes \(\Delta=\mathrm{Disc}(f)\). The curve is smooth (no singularities) iff \(\Delta\neq 0\); for monic \(f\) this reduces to \(\Delta=\prod_{i<j}(r_i-r_j)^2\).

Detailed Explanation

The Discriminant of a Hyperelliptic Curve \(y^2 = f(x)\)

A hyperelliptic curve of genus \(g\) is typically given by an affine equation of the form \(y^2 = f(x)\). The discriminant is a fundamental invariant used to determine if the curve is smooth or singular. Below is the step-by-step explanation of how it is defined and calculated.

Step 1 – Identify the polynomial \(f(x)\):

The curve is defined by the equation \(y^2 = f(x)\), where \(f(x)\) is a polynomial of degree \(n\). For a curve of genus \(g\), the degree is usually \(n = 2g + 1\) or \(n = 2g + 2\). Let the polynomial be expressed as:

\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \]
Step 2 – Understand the relationship to the polynomial discriminant:

The discriminant of the hyperelliptic curve, often denoted as \(\Delta\), is directly related to the discriminant of the polynomial \(f(x)\). The polynomial discriminant \(\operatorname{Disc}(f)\) measures the separation of the roots of \(f(x)\).
Step 3 – Use the formula for the polynomial discriminant:

If the roots of \(f(x)\) are \(r_1, r_2, \dots, r_n\), the discriminant is defined as:

\[ \operatorname{Disc}(f) = a_n^{2n-2} \prod_{1 \le i < j \le n} (r_i – r_j)^2 \]

In many standard contexts where \(f(x)\) is monic (\(a_n = 1\)), this simplifies to the product of the squared differences of the roots.
Step 4 – Relate the discriminant to smoothness:

The discriminant \(\Delta\) is used as a smoothness criterion. A hyperelliptic curve \(y^2 = f(x)\) is non-singular (smooth) if and only if the discriminant is non-zero:

\[ \Delta \neq 0 \]

If \(\Delta = 0\), it implies that \(f(x)\) has at least one repeated root, which corresponds to a singular point on the affine model of the curve.
Step 5 – Alternative calculation using the Resultant:

In practice, the discriminant can be calculated without finding the roots by using the resultant of the polynomial and its derivative \(f'(x)\):

\[ \operatorname{Disc}(f) = \frac{(-1)^{n(n-1)/2}}{a_n}\,\operatorname{Res}(f,f’) \]
Final Result:

For the hyperelliptic curve \(y^2 = f(x)\), the discriminant is defined as the discriminant of the polynomial \(f(x)\):

\[ \Delta = \operatorname{Disc}(f) \]

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FAQs

What is the discriminant of the hyperelliptic curve \(y^2 = f(x)\)?

The discriminant of the curve is (up to conventional normalization) the polynomial discriminant of \(f\). It vanishes exactly when \(f\) has a repeated root, so it detects affine singularities of the model \(y^2=f(x)\).

How is the polynomial discriminant of \(f\) expressed in terms of its roots?

If \(\deg f = n\) and \(f(x) = a_n \prod_{i=1}^n (x-r_i)\), then \(\operatorname{Disc}(f) = a_n^{2n-2} \prod_{i

How can I compute the discriminant using resultants?

Use the resultant of \(f\) and \(f'\): \(\operatorname{Disc}(f) = \frac{\operatorname{Res}(f,f')}{a_n}\). Computer algebra systems (Sage, Magma, PARI/GP, sympy) compute Res and Disc directly.

When is the hyperelliptic curve smooth?

The affine model \(y^2=f(x)\) is smooth iff \(\operatorname{Disc}(f)\neq 0\). repeated root \(r\) gives the singular point \((r,0)\). (One must also check the point(s) at infinity for completeness in projective models.)

How does the discriminant change under the linear change \(x\mapsto ax+b\)?

Under \(x\mapsto ax+b\) the discriminant scales by \(a^{n(n-1)}\) where \(n=\deg f\). Translation by \(b\) does not change the value other than this scaling effect from \(a\).

What happens to the discriminant when you twist by a scalar \(c\): \(y^2 = c\cdot f(x)\)?

How does the degree of \(f\) determine the genus?

If \(\deg f=n\) and \(f\) is separable, the genus is \(g=\lfloor (n-1)/2\rfloor\). Typical models use \(n=2g+1\) or \(n=2g+2\).

What does vanishing of the discriminant modulo a prime \(p\) mean for reduction?

If \(\operatorname{Disc}(f)\equiv 0\pmod p\) the reduction mod \(p\) of the given model is singular: the curve has bad reduction at \(p\). The valuation \(v_p(\operatorname{Disc}(f))\) helps measure severity; a minimal discriminant refines this.

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