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Chapter 21.2 of Friedlander and Iwaniec's Opera de Cribro begins (essentially) as follows:

Let $$F(X, Y) = aX^2 + bXY + cY^2 + \alpha X + \beta Y + \gamma \in \mathbb{Z}[X, Y]$$ be irreducible in $\mathbb{Q}[X, Y]$, represent arbitrarily large odd integers, and satisfy $(a, b, c, \alpha, \beta, \gamma) = 1$. We want $F(X, Y)$ to depend essentially on two variables, which may be expressed by the requirement that $\frac{\partial F}{\partial X}, \frac{\partial F}{\partial Y}$ are linearly independent.

We introduce two discriminants: $$\Delta = b^2 - 4 ac,$$ $$D = a \beta^2 - b \alpha \beta + c \alpha^2 + \Delta \gamma.$$

[end of excerpt]

Wait, they do what?

F+I go on to state that any $F(X, Y)$ satisfying the above conditions represents infinitely many primes: on the order of $\frac{x}{\log x}$ primes $\leq x$ if $D = 0$ or $\Delta$ is a square, and on the order of $\frac{x}{(\log x)^{3/2}}$ otherwise.

They say only a very little bit about their proof, referring instead to this paper of Iwaniec. Obviously these discriminants are important (Iwaniec calls them the "small" and "large" discriminants respectively), but Iwaniec does not explain them in any highbrow fashion; he simply dives into the guts of some computations. Predictably, they are related to affine changes of coordinates (we want to say that $F, F'$ are equivalent if an invertible such transformation changes one into the other), but if the role of $D$ is something simple then this wasn't apparent from a quick look at the paper.

Are these two discriminants classical and familiar? Is there a simple highbrow explanation of what they say about the polynomial $F$? It seems there must be.

(See also here for a related MO question.)

Thank you!

Chapter 21.2 of Friedlander and Iwaniec's Opera de Cribro begins (essentially) as follows:

Let $$F(X, Y) = aX^2 + bXY + cY^2 + \alpha X + \beta Y + \gamma \in \mathbb{Z}[X, Y]$$ be irreducible in $\mathbb{Q}[X, Y]$, represent arbitrarily large odd integers, and satisfy $(a, b, c, \alpha, \beta, \gamma) = 1$. We want $F(X, Y)$ to depend essentially on two variables, which may be expressed by the requirement that $\frac{\partial F}{\partial X}, \frac{\partial F}{\partial Y}$ are linearly independent.

We introduce two discriminants: $$\Delta = b^2 - 4 ac,$$ $$D = a \beta^2 - b \alpha \beta + c \alpha^2 + \Delta \gamma.$$

[end of excerpt]

Wait, they do what?

F+I go on to state that any $F(X, Y)$ satisfying the above conditions represents infinitely many primes: on the order of $\frac{x}{\log x}$ primes $\leq x$ if $D = 0$ or $\Delta$ is a square, and on the order of $\frac{x}{(\log x)^{3/2}}$ otherwise.

They say only a very little bit about their proof, referring instead to this paper of Iwaniec. Obviously these discriminants are important (Iwaniec calls them the "small" and "large" discriminants respectively), but Iwaniec does not explain them in any highbrow fashion; he simply dives into the guts of some computations. Predictably, they are related to affine changes of coordinates (we want to say that $F, F'$ are equivalent if an invertible such transformation changes one into the other), but if the role of $D$ is something simple then this wasn't apparent from a quick look at the paper.

Are these two discriminants classical and familiar? Is there a simple highbrow explanation of what they say about the polynomial $F$? It seems there must be.

(See also here for a related MO question.)

Thank you!

Chapter 21.2 of Friedlander and Iwaniec's Opera de Cribro begins (essentially) as follows:

Let $$F(X, Y) = aX^2 + bXY + cY^2 + \alpha X + \beta Y + \gamma \in \mathbb{Z}[X, Y]$$ be irreducible in $\mathbb{Q}[X, Y]$, represent arbitrarily large odd integers, and satisfy $(a, b, c, \alpha, \beta, \gamma) = 1$. We want $F(X, Y)$ to depend essentially on two variables, which may be expressed by the requirement that $\frac{\partial F}{\partial X}, \frac{\partial F}{\partial Y}$ are linearly independent.

We introduce two discriminants: $$\Delta = b^2 - 4 ac,$$ $$D = a \beta^2 - b \alpha \beta + c \alpha^2 - \Delta \gamma.$$

[end of excerpt]

Wait, they do what?

F+I go on to state that any $F(X, Y)$ satisfying the above conditions represents infinitely many primes: on the order of $\frac{x}{\log x}$ primes $\leq x$ if $D = 0$ or $\Delta$ is a square, and on the order of $\frac{x}{(\log x)^{3/2}}$ otherwise.

They say only a very little bit about their proof, referring instead to this paper of Iwaniec. Obviously these discriminants are important (Iwaniec calls them the "small" and "large" discriminants respectively), but Iwaniec does not explain them in any highbrow fashion; he simply dives into the guts of some computations. Predictably, they are related to affine changes of coordinates (we want to say that $F, F'$ are equivalent if an invertible such transformation changes one into the other), but if the role of $D$ is something simple then this wasn't apparent from a quick look at the paper.

Are these two discriminants classical and familiar? Is there a simple highbrow explanation of what they say about the polynomial $F$? It seems there must be.

(See also here for a related MO question.)

Thank you!

Chapter 21.2 of Friedlander and Iwaniec's Opera de Cribro begins (essentially) as follows:

Let $$F(X, Y) = aX^2 + bXY + cY^2 + \alpha X + \beta Y + \gamma \in \mathbb{Z}[X, Y]$$ be irreducible in $\mathbb{Q}[X, Y]$, represent arbitrarily large odd integers, and satisfy $(a, b, c, \alpha, \beta, \gamma) = 1$. We want $F(X, Y)$ to depend essentially on two variables, which may be expressed by the requirement that $\frac{\partial F}{\partial X}, \frac{\partial F}{\partial Y}$ are linearly independent.

We introduce two discriminants: $$\Delta = b^2 - 4 ac,$$ $$D = a \beta^2 - b \alpha \beta + c \alpha^2 + \Delta \gamma.$$

[end of excerpt]

Wait, they do what?

F+I go on to state that any $F(X, Y)$ satisfying the above conditions represents infinitely many primes: on the order of $\frac{x}{\log x}$ primes $\leq x$ if $D = 0$ or $\Delta$ is a square, and on the order of $\frac{x}{(\log x)^{3/2}}$ otherwise.

They say only a very little bit about their proof, referring instead to this paper of Iwaniec. Obviously these discriminants are important (Iwaniec calls them the "small" and "large" discriminants respectively), but Iwaniec does not explain them in any highbrow fashion; he simply dives into the guts of some computations. Predictably, they are related to affine changes of coordinates (we want to say that $F, F'$ are equivalent if an invertible such transformation changes one into the other), but if the role of $D$ is something simple then this wasn't apparent from a quick look at the paper.

Are these two discriminants classical and familiar? Is there a simple highbrow explanation of what they say about the polynomial $F$? It seems there must be.

(See also here for a related MO question.)

Thank you!

Chapter 21.2 of Friedlander and Iwaniec's Opera de Cribro begins (essentially) as follows:

Let $$F(X, Y) = aX^2 + bXY + cY^2 + \alpha X + \beta Y + \gamma \in \mathbb{Z}[X, Y]$$ be irreducible in $\mathbb{Q}[X, Y]$, represent arbitrarily large odd integers, and satisfy $(a, b, c, \alpha, \beta, \gamma) = 1$. We want $F(X, Y)$ to depend essentially on two variables, which may be expressed by the requirement that $\frac{\partial F}{\partial X}, \frac{\partial F}{\partial Y}$ are linearly independent.

We introduce two discriminants: $$\Delta = b^2 - 4 ac,$$ $$D = \alpha \beta^2 - b \alpha \beta + c \alpha^2 - \Delta \gamma.$$

[end of excerpt]

Wait, they do what?

F+I go on to state that any $F(X, Y)$ satisfying the above conditions represents infinitely many primes: on the order of $\frac{x}{\log x}$ primes $\leq x$ if $D = 0$ or $\Delta$ is a square, and on the order of $\frac{x}{(\log x)^{3/2}}$ otherwise.

They say only a very little bit about their proof, referring instead to this paper of Iwaniec. Obviously these discriminants are important (Iwaniec calls them the "small" and "large" discriminants respectively), but Iwaniec does not explain them in any highbrow fashion; he simply dives into the guts of some computations. Predictably, they are related to affine changes of coordinates (we want to say that $F, F'$ are equivalent if an invertible such transformation changes one into the other), but if the role of $D$ is something simple then this wasn't apparent from a quick look at the paper.

Are these two discriminants classical and familiar? Is there a simple highbrow explanation of what they say about the polynomial $F$? It seems there must be.

(See also here for a related MO question.)

Thank you!

Chapter 21.2 of Friedlander and Iwaniec's Opera de Cribro begins (essentially) as follows:

Let $$F(X, Y) = aX^2 + bXY + cY^2 + \alpha X + \beta Y + \gamma \in \mathbb{Z}[X, Y]$$ be irreducible in $\mathbb{Q}[X, Y]$, represent arbitrarily large odd integers, and satisfy $(a, b, c, \alpha, \beta, \gamma) = 1$. We want $F(X, Y)$ to depend essentially on two variables, which may be expressed by the requirement that $\frac{\partial F}{\partial X}, \frac{\partial F}{\partial Y}$ are linearly independent.

We introduce two discriminants: $$\Delta = b^2 - 4 ac,$$ $$D = a \beta^2 - b \alpha \beta + c \alpha^2 - \Delta \gamma.$$

[end of excerpt]

Wait, they do what?

F+I go on to state that any $F(X, Y)$ satisfying the above conditions represents infinitely many primes: on the order of $\frac{x}{\log x}$ primes $\leq x$ if $D = 0$ or $\Delta$ is a square, and on the order of $\frac{x}{(\log x)^{3/2}}$ otherwise.

They say only a very little bit about their proof, referring instead to this paper of Iwaniec. Obviously these discriminants are important (Iwaniec calls them the "small" and "large" discriminants respectively), but Iwaniec does not explain them in any highbrow fashion; he simply dives into the guts of some computations. Predictably, they are related to affine changes of coordinates (we want to say that $F, F'$ are equivalent if an invertible such transformation changes one into the other), but if the role of $D$ is something simple then this wasn't apparent from a quick look at the paper.

Are these two discriminants classical and familiar? Is there a simple highbrow explanation of what they say about the polynomial $F$? It seems there must be.

(See also here for a related MO question.)

Thank you!