On the Rationalizability of Square Roots in Two Variables

When computing Feynman integrals in theoretical particle physics, one often encounters expressions involving square roots of polynomial functions in multiple variables. A central question is whether these square roots can be "rationalized," that is, transformed into expressions involving only rational functions after a suitable change of variables. Rationalizability is closely tied to the geometric properties of certain algebraic varieties. In the Rationalizability of square roots paper, Dr. Dino Festi and I develop various criteria and computational approaches for determining whether given sets of square roots are rationalizable.

One of the main results we present is the NonADE function for the Computational Algebra System Magma. NonADE can be used to decide rationalizability for square roots in two variables by checking whether the associated double cover of a given square root has only ADE singularities. This post provides a concise guide on how to use NonADE within the Magma Online Calculator, and illustrates these steps with a concrete example from the paper.

How to Set Up NonADE in the Magma Online Calculator

To use the NonADE function in the Magma Online Calculator, first copy the contents of the NonADE implementation and paste the code into the calculator.

You can then call the function below its definition via

NonADE(basering, polynomial);

where polynomial is the defining polynomial of the branch curve and basering is the ring $k[X_1,...,X_n]$ containing that polynomial. The output will list all non-ADE singularities that the double cover has over $k$. If the double cover is smooth or has only ADE singularities, the function will return the empty list.

A Real-World Application of NonADE

The Square Roots in Question

To see how NonADE is to be applied in practice, consider the following alphabet, encountered in perturbative corrections to di-photon and di-jet hadro-production:

We write $f_1,..., f_5$ for the square root arguments. To show that $\mathcal{A}$ is not rationalizable, we consider $J=\{1, 2, 3, 4, 5\}$, define

and denote the associated double cover of $\sqrt{f}$ by $\overline{S}$. In other words, $\overline{S}$ is the hypersurface in the weighted projective space $\mathbb{P}_\mathbb{C}(1,1,1,3)$ defined by the equation

The branch curve of $\overline{S}$ is, therefore, given by the projective curve $\overline{B}\subset\mathbb{P}^2_{\mathbb{C}}$, defined as the zeros of the polynomial

Since $\text{deg}(f)=\text{deg}(F)=6$, we can prove that $\mathcal{A}$ is not simultaneously rationalizable by showing that $\overline{S}$ has at most ADE singularities.

Taking Care of Irrationalities

Before we can use NonADE, though, there is one important subtlety that we have to address first: not all singular points of $\overline{S}$ have rational coordinates. This poses a challenge because, although Magma permits selecting $k=\mathbb{Q}$ as the base field, it does not allow choosing $k=\mathbb{C}$. That means, if we would determine the singular points of $\overline{S}$ over $k=\mathbb{Q}$, NonADE would only classify the singular points of $\overline{S}$ that have purely rational coordinates and would, therefore, miss two of the singular points that $\overline{S}$ has over the complex numbers.

We can resolve this problem by first computing the singular points with a different computer algebra software that will not only be sensitive to singularities over $\mathbb{Q}$ but to all singular points of $\overline{S}$ over $\mathbb{C}$. Since all singularities of a double cover stem from its branch locus, it suffices to look at the singularities of $\overline{B}$. To determine the singular points of $\overline{B}$, one can use the Jacobi criterion, which says that $p$ is a singular point of $\overline{B}$ if and only if

For example, we can compute the singular points in Mathematica via

F:= (x+z)*(x-z)*(y+z)*(x+y+z)*(16*x*z+(4*z+y)^2)

Solve[F==0 && D[F,x]==0 && D[F,y]==0 && D[F,z]==0]

giving us the output

{{x -> -z, y -> 0}, {x -> -z, y -> -8 z}, {x -> -z, y -> -z},
 {x -> z, y -> (-4 - 4i) z}, {x -> z, y -> (-4 + 4i) z},
 {x -> z, y -> -2 z}, {x -> z, y -> -z},
 {x -> -((9 z)/16), y -> -z}, {x -> -9 z, y -> 8 z},
 {x -> 0, z -> 0}, {y -> 0, z -> 0}, {x -> 0, y -> -z},
 {x -> 0, y -> 0, z -> 0}}

Notice that this output should be interpreted as a list of points in $\mathbb{P}_\mathbb{C}^2$, and that we ignore the trivial solution since $(0:0:0)$ does not define a point in the projective plane. We see that $\overline{B}$ has twelve different singular points as a curve over $\mathbb{C}$. Furthermore, we note that two points have an irrational number in their coordinates: the complex unit $i$. Therefore, the coordinates of the two corresponding singularities of $\overline{S}$ will also involve the complex unit.

Applying NonADE to Decide Rationalizability

Knowing which irrationalities occur, we can perform the remaining analysis in Magma: to ensure that NonADE takes all singular points of $\overline{S}$ into account, we have to adjoin $i$ to the coefficient field $\mathbb{Q}$ of our base ring. Put differently, we have to pass from $\mathbb{Q}$ to the extension field $\mathbb{Q}(\sqrt{-1})$.

A convenient way to construct an extension field for $\mathbb{Q}$ is to consider a quotient ring of the polynomial ring $\mathbb{Q}[q]$ that corresponds to the irrational numbers we want to be contained in the extension field. In our example, we want to extend $\mathbb{Q}$ by $i$, i.e., we want to adjoin an element $i$ that satisfies $i^2+1=0$. The field $\mathbb{Q}(\sqrt{-1})$ is, therefore, isomorphic to the quotient $\mathbb{Q}[q]/(q^2+1)$. In Magma, we can define $\mathbb{Q}(\sqrt{-1})$ via

QQ:=Rationals();

E<i>:=ext<QQ|[Polynomial([1,0,1])]>; 

where Polynomial([1,0,1]) specifies the coefficients of the polynomial $g(q)$ in the quotient $\mathbb{Q}[q]/g$, i.e., $g=1\cdot q^0+0\cdot q^1+1\cdot q^2$ in our case.

Now, we can easily prove the non-rationalizability of $\mathcal{A}$ using NonADE:

/////////////////////////////////////////
// paste implementation of NonADE here //
/////////////////////////////////////////

k:=Rationals();

E<i>:=ext<k|[Polynomial([1,0,1])]>;

R<x,y,z>:=PolynomialRing(E,3);

F:=(x+z)*(x-z)*(y+z)*(x+y+z)*(16*x*z+(4*z+y)^2);

NonADE(R,F);

// Output: {[]}

Since NonADE returns the empty list, we conclude that $\overline{S}$ has at most ADE singularities. As a result, $\mathcal{A}$ is not rationalizable.

Summary

In this post, we discussed the connection between the rationalizability of square roots in two variables and the classification of singularities on associated double covers. We introduced the NonADE function for Magma, which identifies whether a given double cover has only ADE singularities. By examining a concrete example from high-energy particle physics, we demonstrated how to handle irrational coordinates, create suitable field extensions, and apply NonADE to decide on the rationalizability of the corresponding square roots.