What can algebraic geometry do for
rational functions in scattering amplitudes?

Giuseppe De Laurentis

CERN - TH

Two-loop leading-color QCD corrections for Higgs plus two-jet production in the heavy-top limit

arXiv:2605.04009 / JHEP07(2026)251 with H. Ita, V. Kuschke, M. Ruf, and V. Sotnikov

Primary Decompositions in Lorentz-Covariant Rings

arXiv:2503.10595 with D. Tai (master student)

Factoriality of massless spinor rings beyond four points

In preparation

Positive Geometry: Back to Physics - MPI Leipzig

$\;$

Find these slides at gdelaurentis.github.io/slides/leipzigoct2026

Part 1:
Computing Amplitudes

Partial Amplitudes & Finite Remainders

$\circ$ Amplitude (integrands) can be written (for a suitable choice of master integrals) as

$$ \displaystyle A(\lambda, \tilde\lambda, \ell) = \sum_{\substack{\Gamma,\\ i \in M_\Gamma \cup S_\Gamma}} \, c_{\,\Gamma,i}(\lambda, \tilde\lambda, \epsilon) \, \frac{m_{\Gamma,i}(\lambda\tilde\lambda, \ell)}{\textstyle \prod_{j} \rho_{\,\Gamma,j}(\lambda\tilde\lambda, \ell)} \;\; \xrightarrow[]{\int d^D\ell} \;\; \sum_{\substack{\Gamma,\\ i \in M_\Gamma}} \frac{ \sum_{k=0}^{\text{finite}} \, {\color{red}c^{(k)}_{\,\Gamma, i}}(\lambda, \tilde\lambda) \, \epsilon^k}{\prod_j (\epsilon - a_{ij})} \, {\color{orange}I_{\Gamma, i}}(\lambda\tilde\lambda, \epsilon) $$

$\circ$ $\Gamma$: topologies $\quad\circ$ $M_\Gamma$: master integrands $\quad\circ$ $S_\Gamma$: surface terms

Abreu, Dormans,

Febres Cordero, Ita

Kraus, Page, Pascual,

Ruf, Sotnikov ('20)

$\circ$ All physical information is contained in the finite remainders, at two loops

Weinzierl ('11)

$$ \underbrace{\mathcal{R}^{(2)}}_{\text{finite remainder}} = \mathcal{A}^{(2)}_R \underbrace{- \quad I^{(1)}\mathcal{A}^{(1)}_R \quad - \quad I^{(2)}\mathcal{A}^{(0)}_R}_{\text{divergent + convention-dependent finite part}} + \mathcal{O}(\epsilon) $$

Catani ('98) Becher, Neubert ('09) Gardi, Magnea ('09)

$\phantom{\circ}$ $\mathcal{A}^{(1)}_R$ to order $\epsilon^2$ is still needed to build $\mathcal{R}^{(2)}$, but there is no real physical reason to reconstruct it.

$\circ$ Finite remainder as a weighted sum of pentagon functions + Zoia ('21) Chicherin, Sotnikov ('20) + Abreu, Ita, Page, Tschernow ('23)

$$ \textstyle \mathcal{R}(\lambda, \tilde\lambda) = \sum_i \color{red}{r_{i}(\lambda,\tilde\lambda)} \, \color{orange}{G_i(\lambda\tilde\lambda)} $$

$\circ$ Goal: reconstruct $\color{red}{r_{i}(\lambda,\tilde\lambda)}$ from numerical samples in a field $\mathbb{F}$

$\mathbb{F}_p$: von Manteuffel, Schabinger ('14); $\phantom{\mathbb{F}_p}$ Peraro ('16)
$\mathbb{C}$: GDL, Maitre ('19); $\mathbb{Q}_p$: GDL, Page ('22)

Variable Quotient Rings

based on work with Ben Page:
arXiv:2203.04269 (JHEP 12 (2022) 140)

related mathematical treatment of spinor-helicity varieties, Sturmfels et al:
arXiv:2406.17331 (SIAM 24M1671840)

Massless Scattering

$\circ$ There are two sources of redundancies: kinematic constraints & Schouten/Gram identities.

$\circ$ For $n$-point massless scattering, the covariant quotient ring is

$$ \displaystyle \kern10mm R_{n} = \mathbb{F}\Big[|1⟩_{\alpha}, [1|_{\dot\alpha}, \dots, |n⟩_{\alpha}, [n|_{\dot\alpha} \Big] \Big/ \Big\langle {\textstyle \sum_{i=1}^n} |i\rangle[ i | \Big\rangle $$

$\circ$ The $4$ four-momentum conservation redundancies yield equivalence relations such as

$$ \displaystyle \langle 1|2+3|1]=\langle 1|-1-4-5|1]=-\langle 1|4+5|1] \quad \text{in} \quad R_5 $$

$\circ$ The rational functions $r_i$ belong to the field of fractions of $R_n$ (or $\mathcal{R}_n \cong R_n^{SL(2)_L\times SL(2)_R}$),

$$ \displaystyle r_i(|i\rangle,[i|) = \frac{\mathcal{N}(|i\rangle,[i|)}{\mathcal{D}(|i\rangle,[i|)} \, , \quad r_i(|i\rangle,[i|) \in \text{Frac}(R_n) $$

$\circ$ Interesting mathematical observations ~~and open questions~~:
$\quad\star$ $R_3$ is not an Integral Domain, i.e. it breaks $ab=0 \Rightarrow a = 0 \text{ or } b = 0$ (zero divisors)
$\quad\star$ $R_4$ is not an Unique Factorization Domain (which is why MHV = anti-MHV)
$\quad\star$ ~~Conjecture~~ Theorem: $R_{n\geq 5}$ is UFD. E.g. this implies uniqueness of reduced denominators
$\phantom{\circ}$ Note: all polynomial rings are UFD, so clearly $R_4$ is not equivalent to one, e.g. $\mathbb{F}[s,t]$

Covariant Q-Rings for Massive Processes

$\circ$ With a single massive leg, e.g. $pp \rightarrow Hjj$, we have $R_{n-1}$ without momentum conservation.

$$ \displaystyle \kern10mm R_{Hjj} = \mathbb{F}\big[|1⟩_{\alpha}, [1|_{\dot\alpha}, |2⟩_{\alpha}, [2|_{\dot\alpha}, |3⟩_{\alpha}, [3|_{\dot\alpha}, |4⟩_{\alpha}, [4|_{\dot\alpha} \big] $$

$\phantom{\circ}$ This does not work for multiple massive legs, due to $p_{M_1} \cdot p_{M_2}$, or with states.

GDL, Ita, Page, Sotnikov

$\circ$ In the massive (or spin-spinor) formalism we have, for example

Shadmi, Weiss Ochirov; Arkani-Hamed, Huang, Huang;

$$ \displaystyle \kern10mm R_{Vjj} = \mathbb{F}\big[|1⟩_{\alpha}, [1|_{\dot\alpha}, |2⟩_{\alpha}, [2|_{\dot\alpha}, |3⟩_{\alpha}, [3|_{\dot\alpha}, |4⟩_{\alpha}, [4|_{\dot\alpha}, |\boldsymbol 5⟩^J_{\alpha}, [\boldsymbol 5|^I_{\dot\alpha} \big] \Big/ \Big\langle {\textstyle \sum_{i=1}^4} |i]\langle i | + |\boldsymbol 5⟩^I_{\alpha}[\boldsymbol 5|_{I,\dot\alpha} \Big\rangle $$

GDL, Melnikov, Tresoldi

$$ \displaystyle \text{with} \qquad p_V = |\boldsymbol q^I\rangle [\boldsymbol q_I| = |\boldsymbol q^1\rangle [\boldsymbol q_1| + |\boldsymbol q^2\rangle [\boldsymbol q_2| = |5\rangle [5| + |6\rangle [6| $$

$$ \displaystyle \varepsilon^{\mu,IJ}_{\boldsymbol 5} = \frac{1}{\sqrt{2}m}[\boldsymbol 5^I|\gamma^\mu|\boldsymbol 5^J\rangle \quad \text{and} \quad \varepsilon^{-} \propto \varepsilon^{11}, \; \varepsilon^{L} \propto \varepsilon^{21} + \varepsilon^{12}, \; \varepsilon^{+} \propto \varepsilon^{22} \quad \text{physical} $$

$\circ$ While $pp \rightarrow ttH$ exposes the full complexity, with multiple massive states

Campbell, GDL, Ellis

$$ \displaystyle \kern10mm R_{ttH} = \frac{\mathbb{F}\big[|1⟩_{\alpha}, [1|_{\dot\alpha}, |2⟩_{\alpha}, [2|_{\dot\alpha}, |\boldsymbol{3}^I⟩_{\alpha}, [\boldsymbol{3}^I|_{\dot\alpha}, |\boldsymbol{4}_J⟩_{\alpha}, [\boldsymbol{4}_J|_{\dot\alpha}, \boldsymbol{5}_{\alpha\dot\alpha} \big]}{\big\langle \sum_{i,I,J} |i\rangle[i|, \langle \boldsymbol{3}|\boldsymbol{3}⟩ +[\boldsymbol{3}|\boldsymbol{3}], \langle \boldsymbol{3}|\boldsymbol{3}⟩-\langle \boldsymbol{4}|\boldsymbol{4}⟩, \langle \boldsymbol{4}|\boldsymbol{4}⟩ +[\boldsymbol{4}|\boldsymbol{4}]\big\rangle} $$

$\phantom{\circ}$ where $\langle \boldsymbol{3}^I|\boldsymbol{3}^J⟩=m\epsilon^{JI} \text{ and } [\boldsymbol{3}^I|\boldsymbol{3}^J]=\bar{m}\epsilon^{IJ}$; we are setting $m=\bar{m}$ and the tops on-shell.

Examples of Trees

$\circ$ To not make this too abstract, we are after expressions like these, but for the MI coefficients.

$\circ$ For $Vjj$ there are 5 amplitudes (showing 3)

$$ {A}_g^{(0)}(1^{+}_\bar{q}, 2^{+}_g, 3^{+}_g, 4^{-}_q, 5^{+}_\bar{\ell}, 6^{-}_\ell) = \frac{⟨46⟩^2}{⟨12⟩⟨23⟩⟨34⟩⟨65⟩} \, \Rightarrow {A}_g^{(0),{IJ}}(1^{+}_\bar{q}, 2^{+}_g, 3^{+}_g, 4^{-}_q, {\boldsymbol 5}) = \frac{⟨4|\boldsymbol 5|\boldsymbol 5^I]⟨\boldsymbol 5^J 4⟩}{⟨12⟩⟨23⟩⟨34⟩s_{\boldsymbol 5}} , \\[6mm] {A}_g^{(0)}(1^{+}_\bar{q}, 2^{+}_g, 3^{-}_g, 4^{-}_q, 5^{+}_\bar{\ell}, 6^{-}_\ell) = \frac{⟨13⟩⟨3|1+2|5]^2}{⟨12⟩⟨23⟩[65]⟨1|2+3|4]s_{123}} \; + \; (123456\rightarrow \overline{432165}) \, , \\[6mm] {A}_q^{(0)}(1^{+}_\bar{q}, 2^{+}_{q'}, 3^{+}_{\bar{q}'}, 4^{-}_q, 5^{+}_\bar{\ell}, 6^{-}_\ell) = -\frac{[12]⟨46⟩⟨3|1+2|5]}{⟨23⟩[23]⟨56⟩[56]s_{123}}+(123456\rightarrow 156423)\phantom{+} $$

$\circ$ For $q\bar{q}\rightarrow t\bar{t}H$ there is only a single amplitude

$$ {A}_{ttH}^{(0)}(1^{+}_q, 2^{-}_\bar{q}, 3_t, 4_\bar{t}, 5_H)^I_J = \frac{⟨2|𝟑|1]⟨𝟑^I𝟒_J⟩-[𝟑^I1][1𝟒_J]⟨12⟩}{s_{12}(s_{12𝟑}-m_t²)} + (12345\rightarrow\overline{21345},12435,\overline{21435}) $$

$\phantom{\circ}$ where for clarity I have not suppressed the spin indices. Symmetries are made manifest.

$\phantom{\circ}$ Note: The amplitude is spin covariant, just like it is little group covariant!
$\phantom{\circ} \kern7mm$ We need only obtain a single choice, say $I=J=1$, the other follows.

Spinor Alphabets

$\circ$ We can always factorize a polynomial into products of irreducible factors, to some powers

$$ \displaystyle r_i(|i\rangle,[i|) = \frac{\mathcal{N}(|i\rangle,[i|)}{\prod_j \mathcal{D}_j^{q_{ij}}(|i\rangle,[i|)} % \, , \quad r_i(|i\rangle,[i|) \in \text{Frac}(R_n) $$

$\phantom{\circ}$ The denominator factors $\mathcal{D}_j$ are conjectured to be (mostly) related to the letters of the symbol alphabet

Abreu, Dormans, Febres Cordero, Ita, Page ('18)

$\circ$ Convert your alphabet from independent Mandelstam invariants to redudant spinors brackets

From work in progress with S. Abreu, X. Liu, P.F. Monni

Mandelstam letters
$s_{12}$
$s_{123}$
$s_{12} - s_{123} - s_{345} + s_{45}$
$-s_{12} + s_{123}$
$s_{12}(s_{123} - s_{56}) - s_{123}(s_{123} + s_{34} - s_{56})$
$\displaystyle\frac{ s_{12}\left(s_{16}(s_{23} - s_{234})s_{34} + s_{23}^{2}(\cdots) + \cdots\right) + s_{123}(\cdots) + s_{23}(\cdots) }{ \sqrt{(-s_{12} + s_{123} - s_{23})^2\cdots} }$

$\Rightarrow$

Spinor letters
$\langle 1\,2\rangle[1\,2]$
$s_{123}$
$\langle 3\,|\,6\rangle[3\,|\,6]$
$\langle 3\,|\,1{+}2\,|\,3]$
$\langle 3\,|\,1{+}2\,|\,4]\langle 4\,|\,1{+}2\,|\,3]$

$\operatorname{tr}_5(2,3,4,5)$

$\circ$ Factorization and extra chiral cancellations are key for simplification in gauge amplitudes

We now have a factorization algorithm for spinors! See Part 2.

Reduced
Denominators

(Geometry at codimension one)

Least Common Denominator

$\circ\,$ We can now determine the least common denominators (LCDs),

$$ \displaystyle \mathcal{D} = \prod_j \mathcal{D}_j^{q_{ij}}(|i\rangle,[i|) \, . $$

$\phantom{\circ}\,$ Obtain the $q_{ij}$ from a univariate slice $\vec\lambda(t)$, i.e. a 1D curve.

$\circ$ The curve must intersect all varieties $V(\langle \mathcal{D}_j \rangle)$, e.g.

$\phantom{\circ}\,$ Solve for $a_i, b_i$ such that constraints are satisfied.

Space has dimension $4n-4$,

$\mathcal{D}_j = 0$ have dimension $4n-5$,

$\vec\lambda(t)$'s have dimension 1.

Poles & Zeros $\;\Leftrightarrow\;$ Irreducible Varieties $\;\Leftrightarrow\;$ Prime Ideals
Physics $\kern18mm$ Geometry $\kern18mm$ Algebra

Open-Source Implementation

$\circ\,$ Implemented in antares, lips, linac (~~coming soon!~~ now open-source!), and syngular.

$\circ\,$ Example: determine the LCDs of a vector of finite-remainder coefficients.

from antares.core.numerical_methods import tensor_function
from lips import Particles

oTF = tensor_function(lambda oPs: numpy.array(
    [coeff for coeff, monom in amps['qbpqmqbpqm_2L_Nc2_Nf0'].finite_remainder(oPs)] +
    [coeff for coeff, monom in amps['qbpqmqbpqm_2L_Nc1_Nf1'].finite_remainder(oPs)] +
    ....
))
oTF.multiplicity = 6
oTF.__name__ = '4q1h_mhv'  # qbar+ q- qbar+ q- H, through two loops

oSlice = Particles(6, field=Field("finite field", 2 ** 31 - 1, 1), seed=0)
oSlice.univariate_slice(algorithm='covariant', seed=0)

lTermsLCD = oTF.get_lcds(oSlice, verbose=True)
lTermsLCD[:5]

[
  Terms("""+(1)/(⟨1|2⟩[1|2]⟨3|4⟩[3|4]⟨1|3+4|1]⟨2|3+4|2]⟨3|1+2|3]⟨4|1+2|4])"""),
  Terms("""+(1)/(⟨1|2⟩[1|2]⟨1|3⟩[2|4]⟨3|4⟩[3|4]⟨1|2+3|1]⟨4|2+3|4])"""),
  Terms("""+(1[1|3])/(⟨1|3⟩⟨1|2+3|1]⟨3|1+2|3]²)"""),
  Terms("""+(1⟨2|4⟩)/([2|4]⟨2|3+4|2]²⟨4|2+3|4])"""),
  ...
]

LCD Factors / Kinematic Poles / Letters

$\circ\,$ The irreducible denominator factors $\mathcal{D}_j \text{ for } Vjj$ (modding out by permutation orbits) read

$$ \displaystyle \mathcal{D}_{Hjj} = \mathcal{D}_{Vjj} \subset \kern-3mm \bigcup_{\sigma \; \in \; \text{Aut}(R_6)} \sigma \circ \big\{ \langle 12 \rangle, \langle 1|2+3|1], \langle 1|2+3|4], s_{123}, \Delta_{12|34|56}, \underbrace{⟨3|2|5+6|4|3]-⟨2|1|5+6|4|2]}_{\normalsize\text{only new one at two loops!}} \big\} $$

$\circ\,$ For $t\bar{t}H$ (at one-loop), they read

$$ \displaystyle \kern-10mm \mathcal{D}_{ttH} = \big\{ \langle 12 \rangle, [12], s_{123}, \dots, (s_{123}-m^2), \langle 1|\boldsymbol{3}|1], \dots, \\[2mm] \kern30mm \langle 1|\boldsymbol{3}|\boldsymbol{4}| 2 \rangle, \dots, \langle 1|\boldsymbol{3}|1+2|\boldsymbol{4}| 2], \dots, \Delta_{12|34|5}, \dots \Delta_{12|3|4|5} \big\} $$

$\phantom{\circ}\,$ note that there is no dependence on the top states (this looks like 3 massive scalars).

$\circ\,$ For $HHH$ (at one-loop), they are

$$ \small \begin{gathered} \mathcal{D}_{HHH} = \big\{ ⟨1|2⟩, [1|2], ⟨2|𝟓|1], ⟨2|𝟒|1], ⟨2|𝟑|1], ⟨1|𝟑|2], [1|𝟑|𝟓|1], ⟨1|𝟑|𝟓|1⟩, ⟨1|𝟓|𝟒|2⟩, [2|𝟒|𝟓|1], Δ_{12|𝟑|𝟒|𝟓}, \\ ⟨2|𝟑|𝟒|𝟓|1], ⟨1|𝟓|𝟒|𝟑|2], ⟨1|2⟩[1|2]⟨1|𝟓|𝟒|𝟑|2]⟨2|𝟑|𝟒|𝟓|1]+m_t^2\text{tr}_5(1|2|𝟑|𝟒)^2, \\ ⟨1|𝟑|2]⟨2|𝟒|𝟓|1⟩[1|𝟑|2⟩[2|𝟒|𝟓|1]+m_t^2\text{tr}_5(1|2|𝟑|𝟒)^2 \big\} \end{gathered} $$

$\circ\,$ Challenge: in LCD form the numerators are intractably complicated.
$\phantom{\circ}\,$ E.g. for $Vjj$ the most complicated function had a mass dim. ($\approx$ poly. degree) of 114 $\Rightarrow$ 25M parameters
$\phantom{\circ}\,\qquad$ (and non-trivial little group weights $\{3, -12, 12, -3, -1, 1\}$ due to chiral cancellations!)
$\phantom{\circ}\,$ For $Hjj$ the most complicated function had a mass dim. ($\approx$ poly. degree) of 168 $\Rightarrow$ 80M parameters

Effect of Basis Change on LCDs

$\circ\,$ Change basis from a subset of the pentagon coefficients $r_{i \in \mathcal{B}}$ to $\mathbb{Q}$-linear combinations $\tilde r$,

$$ R = r_j h_j = r_{i\in \mathcal{B}} M_{ij} h_j = \tilde{r}_{i} \, O_{ii'}M_{i'j} \, h_j \, , \qquad O_{ii'}, M_{i'j}\in \mathbb{Q} $$

[6] Abreu, Febres Cordero, Ita, Klinkert, Page, Sotnikov '21

$\circ\,$ For the recent Hjj computation

Basis Change from Laurent Coefficients

$\circ\,$ By Gaussian elimination, partition the space (abusing notation for residue):

$$ \text{span}(r_{i \in \mathcal{B}}) = \underbrace{\text{column}(\text{Res}(r_{i \in \mathcal{B}}, \mathcal{D}_k^m))}_{\text{functions with the singularity}} \;\;\; \oplus \, \underbrace{\text{null}(\text{Res}(r_{i \in \mathcal{B}}, \mathcal{D}_k^m))}_{\text{functions without the singularity}} $$

$\circ\,$ Search for linear combinations that remove as many singularities as possible (while not dropping rank)

$\phantom{\circ}\,$ $N^{(\text{pole order})}_{\text{denominator factor}}$ denote null-spaces, and arrows denote intersections. Greated observed breadth $\mathcal{O}(100k)$.

Multivariate Partial Fraction Decomposition

(Geometry beyond codimension one)

mPFD as Ideal Membership

GDL, Maître ('19) GDL, Page ('22)

$\circ$ We want a mathematically rigorous answer to the question

$$ \frac{\mathcal{N}}{\mathcal{D}_1\mathcal{D}_2} \stackrel{?}{=} \frac{\mathcal{N}_2}{\mathcal{D}_1} + \frac{\mathcal{N}_1}{\mathcal{D}_2} $$

$\phantom{\circ}$ without knowing $\mathcal{N}$ analytically. The complexity should not depend on $\mathcal{N}$ (besided numerical evaluations).
$\phantom{\circ}$ The complexity will depend on $\mathcal{D}_1, \mathcal{D}_2$

$\circ$ Multivariate partial fraction decompositions follow from varieties where pairs of denominator factors vanish

$$ \frac{\mathcal{N}}{\mathcal{D}_1\mathcal{D}_2} \stackrel{?}{=} \frac{\mathcal{N}_2}{\mathcal{D}_1} + \frac{\mathcal{N}_1}{\mathcal{D}_2} \; \Longleftrightarrow \; \mathcal{N} \stackrel{?}{\in} \big\langle \mathcal{D}_1, \mathcal{D}_2 \big\rangle \, \text{ i.e. } \; \mathcal{N} \stackrel{?}{=} \mathcal{N}_1 \mathcal{D}_1 + \mathcal{N}_2 \mathcal{D}_2 $$

$\cap$

$=$

$$ \langle xy^2 + y^3 - z^2 \rangle + \langle x^3 + y^3 - z^2 \rangle = \langle xy^2 + y^3 - z^2, x^3 + y^3 - z^2 \rangle = \langle 2y^3-z^2, x-y \rangle \cap \langle y^3-z^2, x \rangle \cap \langle z^2, x+y \rangle $$

$\phantom{\circ}$ This is a primary decomposition. If $\mathcal{N}$ vanishes on all branches, than the partial fraction decomposition exists.

Lorentz-Covariant Primary Decompositions

GDL, Tai (‘26)

$\circ$ Generic algorithms (e.g. primdecGTZ) often fail to return in reasonable time.

$\circ$ Gröbner bases and saturations can destroy manifest covariance of the returned generators. E.g.

$$ \langle f_1, f_2\rangle = \langle f_1, \tilde f_2\rangle \quad \text{with} \quad f_2=\widetilde f_2+qf_1 , \qquad \text{take} \quad X\in V(\langle f_1\rangle) \quad\Longrightarrow\quad f_2(X)=\widetilde f_2(X). $$

$\phantom\circ$ We can therefore re-fit a covariant representative on this codimension-one variety:

$$ \widetilde f_2=\sum_j c_j m_j, \qquad A\,c=b, \qquad A_{ij}=m_j(X_i),\quad b_i=f_2(X_i). $$

$\circ$ Semi-numerical determination of independent sets and variety dimensions for $J$ and $V(J)$:

$$ Y\subseteq X\ \text{independent} \quad\Longleftrightarrow\quad J\cap\mathbb{F}[Y]=\{0\}, \qquad |Y|=\dim V(J)\ \text{if $Y$ is maximal}. $$

$\phantom\circ$ Specialise a random subset $W\subseteq X$. If $W$ is contained in an independent set $Y$, then for the sliced ideal $J'$,

$$ \dim V(J)=\dim V(J')+|W|. $$

$\phantom\circ$ Repeated $\mathbb{F}_p$ sampling quickly finds independent sets and determines dimensions, avoiding expensive symbolic elimination.

$\circ$ These tools substantially accelerate primality / primaryness tests for complicated unmixed ideals, as well as point generation near varieties.

Bivariate Slice mPFD Algorithm

A partial work-around

$\circ$ Issues: 1) get prim. dec.; 2) sample at codim. two; 3) still need to intersect ideals

$\circ$ Build a bivariate slice (curve) that intersects all varieties $V(\langle \mathcal{D}_i, \mathcal{D}_j \rangle)$, e.g.

$\phantom{\circ}$ Such that for all $t_1, t_2$ momentum is conserved: gives 20 eqs. defining a codim.-10 variety in 24 dims
$\phantom{\circ}$ Pick sol. with syngular.Ideal.points_on_variety, impl. see lips.Particles.bivariate_slice

$\circ$ Reconstruct (Newton-interpolate) the numerators $\mathcal{N}(t_1, t_2)$; this requires $\text{deg}(\mathcal{N}+1)^2$ points.

$\circ$ For each $\mathcal{J}_{ij}^{a_i,a_j} = \big\langle \mathcal{D}_i(t_1, t_2)^{a_i}, \mathcal{D}_j(t_1, t_2)^{a_j} \big\rangle\;$ test $\;\mathcal{N}_i(t_1, t_2) \in \mathcal{J}_{ij}^{a_i,a_j}$

$\circ$ If the test passes, construct the simplified intersection from least common multiples (lcm's)

$$ \begin{equation} \begin{aligned}\label{eq:RS_merge_generalised} \mathcal{I}_{\rm s}'&= \mathcal{I}_{\rm s} \cap_{s} \mathcal{J}_{ij}^{a_ia_j}:= \big\langle \mathrm{lcm}(m_1,n_1), \mathrm{lcm}(m_1,n_2), \ldots, \mathrm{lcm}(m_r,n_1), \mathrm{lcm}(m_r,n_2) \big\rangle \\ & \kern45mm \mbox{with}\quad n_1=\mathcal{D}_i(u,v)^{a_i} \quad\mbox{and}\quad n_2=\mathcal{D}_j(u,v)^{a_j}\,, \end{aligned} \end{equation} $$

$\phantom{\circ}$ Where $\mathcal{I}_{\rm s}$ starts as the unit ideal, and is otherwise generated by $m_1, \dots, m_r$.

$\circ$ Test $\;\mathcal{N}_i(t_1, t_2) \in \mathcal{I}_{\rm s}'$, if it passes set $\; \mathcal{I}_{\rm s} = \mathcal{I}_{\rm s}'$

Bivariate Slice mPFD Interpretation

$\circ$ In the end we are left with an ideal

$$ \begin{equation} \mathcal{I}_{\rm s}=\langle m_1(\mathcal{D}),\ldots,m_r(\mathcal{D})\rangle \quad \mbox{with} \quad \mathcal{N} \in \mathcal{I}_{\rm s} \,. \end{equation} $$

$\phantom{\circ}$ whose genreators $m_k$ are monomials in the denominator factors $\mathcal{D}_i$,

$$ \begin{align} m_k(\mathcal{D}) = \prod_j \mathcal{D}_j^{\hat q_{jk}} \in \mathcal{I}_{\rm s} \quad\mbox{with}\quad k=1,\ldots, r \end{align} $$

$\phantom{\circ}$ with powers not exceeding those of the LCD, i.e. no spurious singularities by construction.

$\circ$ We interpret this back in the full multivariate setting as

$$ \mathcal{N} = \sum_{k=1}^{k_{\rm max}} \mathcal{N}_k \, m_k(\mathcal{D} ) \, $$

$\phantom{\circ}$ and thus the (tentative) multivariate decomposition (Question: how to detect failures before fitting?)

$$ \begin{equation} \tilde r = \frac{\mathcal{N}}{\prod_j \mathcal{D}_j^{q_j}} = \sum_{k=1}^{r} \frac{\mathcal{N}_k }{\prod_j \mathcal{D}_j^{q_{jk}}} \quad\mbox{with} \quad \prod_j \mathcal{D}_j^{q_{jk}} = \frac{\prod_j \mathcal{D}_j^{q_{j}}}{m_k(\mathcal{D})} \, . \end{equation} $$

$\circ$ Verify by fitting the ansatz. If it fails, remove some ideals from the intersection.

Invariant Sub-Rings for Ansätze

$\circ$ Helicity amplitudes are Lorentz invariant: minimal ansätze are build in the invariant sub-rings.

$\circ$ General construction for Lorentz-Invariant sub-rings through elimination theory

$\quad\star$ Build a ring with both covariant and invariant variables

$$ \mathbb{F}\big[ |i\rangle, [i|, \langle i j\rangle , [ij] \big] $$

$\quad\star$ Define relations among variables (on top of existing constraints)

$$ \big\{ \langle ij \rangle - \epsilon^{\beta\alpha} \lambda_{i\alpha} \lambda_{j, \beta}, [ij] - \tilde\lambda_{i\dot\alpha} \epsilon^{\dot\alpha\dot\beta} \tilde\lambda_{j, \dot\beta} \big\} $$

$\quad\star$ Compute a lexicographical Groebner basis with invariants > covariants

$\circ$ We obtain the following invariant rings

$$ \displaystyle \mathcal{R}_{Vjj} = \frac{\mathbb{F}\big[ \langle ij\rangle : \, 1\leq i< j\leq 6, i,j \neq 5, \; [ij] : 1\leq i< j\leq 5 \big]}{\big\langle {\textstyle \sum_{i=1}^4} [5|i]\langle i |6\rangle, 34 \text{ Schouten identities} \big\rangle} $$

$$ \displaystyle \mathcal{R}_{ttH} = \mathbb{F}\big[ \underbrace{\langle 12\rangle, \langle \boldsymbol{3}1\rangle ... ⟨2|\boldsymbol{3}|2] ... ⟨2|\boldsymbol{3}|\boldsymbol{4}|2⟩}_{37\; \text{invariants}} \big]\Big/ \big\langle \underbrace{⟨2|\boldsymbol{3}|2]⟨2|\boldsymbol{4}|1]-⟨2|\boldsymbol{3}|1]⟨2|\boldsymbol{4}|2]-[1|2]⟨2|\boldsymbol{3}|\boldsymbol{4}|2⟩, ...}_{\text{more than} \; 90 \; \text{generators}} \big\rangle $$

$\circ$ Ansatze built via Groebner bases, and fitted with linear algebra (e.g. $\mathbb{F}_p$ on GPU, see Linac)

Part 2:
Proof of Factoriality

based on upcoming work:
Factoriality of massless spinor rings beyond four points
with independent Lean verification

Factoriality of Massless Spinor Rings

$\circ$ Throughout we have referred to the irreducible denominator factors $\mathcal{D}_i$.
$\phantom{\circ}$ But is the factorization $\mathcal{D}=\prod_i \mathcal{D}_i$ actually unique? i.e. is $R_n$ factorial?

$\circ$ $R_3$ is not even a domain: it has zero divisors,

$$ s_{12}=\langle12\rangle[21]=s_3=0, \qquad \langle12\rangle\neq0,\quad [21]\neq0 . $$

$\circ$ $R_4$ is a domain: the momentum-cons. ideal $J_4=\left\langle\sum_i |i\rangle[i|\right\rangle$ is prime, but $R_4$ is not factorial,

$$ \langle12\rangle[12]=\langle34\rangle[34]. $$

$\phantom{\circ}$ Shows two inequivalent factorizations into non-associate irreducibles.
$\phantom{\circ}$ $\Rightarrow$ 4 point reduced denominators need not be unique: e.g. Parke-Taylor MHV = anti-MHV.

Theorem: $R_n$ is a unique factorization domain for every $n\geq5$.

Integrality for $n\geq4$

$\circ$ Criterion: if a complete-intersection variety contains an integral open subset whose complement has smaller dimension, then the whole coordinate ring is a domain.

$$ R_n= \mathbb F[\Lambda,\widetilde\Lambda]/ \langle\Lambda\widetilde\Lambda\rangle , \qquad \Lambda= \begin{pmatrix} a_1&\cdots&a_n\\ b_1&\cdots&b_n \end{pmatrix}, \qquad \widetilde\Lambda= \begin{pmatrix} c_1&d_1\\ \vdots&\vdots\\ c_n&d_n \end{pmatrix}. $$

$\circ$ Stratify by $r=\operatorname{rank}\Lambda$. A $2\times n$ matrix of rank $r$ depends on $r(n+2-r)$ parameters, while $\dim\ker\Lambda=n-r$, thus

$$ \begin{array}{c|c} r & \dim\{(\Lambda,\widetilde\Lambda):\Lambda\widetilde\Lambda=0\}\\ \hline 2 & 2n+2(n-2)=4n-4\\ 1 & (n+1)+2(n-1)=3n-1\\ 0 & 2n \end{array} $$

$\circ$ Why is the rank-two locus integral? On a chart where $\Delta=\langle12\rangle\neq0$, we can solve

$$ \underbrace{ \begin{pmatrix} a_1&a_2\\ b_1&b_2 \end{pmatrix} }_{M\ {\rm invertible}} \begin{pmatrix} c_1\\ c_2 \end{pmatrix} = -\sum_{i=3}^n c_i \begin{pmatrix} a_i\\ b_i \end{pmatrix}. $$

$3n-1 < 4n-4 \iff n\geq4 \;\; \Longrightarrow \; R_{n\geq 4}$ is a domain.

Choosing a Prime Divisor

$\circ$ For the next step we need a prime element, $\Delta$. Say,

$$ \Delta=\langle12\rangle=a_1b_2-a_2b_1. $$

$\circ$ We study the quotient $R_n/(\Delta)$, i.e. the momentum-conservation variety restricted to $\langle12\rangle=0$.

$\circ$ On the full-rank locus, imposing one additional equation reduces the dimension by one:

$$ \dim(r=2,\Delta=0)=4n-5. $$

$\circ$ The largest lower-rank stratum is $r=1$, which already implies $\Delta=0$,
$\phantom\circ$ and thus still has dimension $3n-1$.

$\circ$ Therefore, by the same argument as before, when

$$ (4n-5)>(3n-1) \quad\Longleftrightarrow\quad n\geq5, $$

$\phantom\circ$ it follows that

$R_{n\geq5}/(\langle12\rangle)$ is a domain $\;\Longrightarrow\; \langle12\rangle$ is prime for $n\geq5$

Nagata's Criterion

$\circ$ For $n\geq5$, we have a Noetherian domain $R_n$ containing the prime element $\Delta=\langle12\rangle$.
$\phantom{\circ}$ We now localize at $\Delta$, i.e. we invert it.

$\circ$ Since $\det M=\Delta\neq0$, the momentum-conservation equations

$$ M \begin{pmatrix} c_1\\ c_2 \end{pmatrix} = -\sum_{i=3}^n c_i \begin{pmatrix} a_i\\ b_i \end{pmatrix}, \qquad M \begin{pmatrix} d_1\\ d_2 \end{pmatrix} = -\sum_{i=3}^n d_i \begin{pmatrix} a_i\\ b_i \end{pmatrix} $$

$\phantom{\circ}$ uniquely solve for $c_1,c_2,d_1,d_2$. Therefore

$$ R_n[\Delta^{-1}] \cong P_n[\Delta^{-1}], \qquad P_n= \mathbb F[ a_1,\ldots,a_n, b_1,\ldots,b_n, c_3,\ldots,c_n, d_3,\ldots,d_n ]. $$

$\circ$ The right-hand side is a localization of a polynomial ring, and is therefore a UFD.

$\circ$ Nagata's criterion:
$\phantom\circ$ Let $B$ be a Noetherian domain and $p\in B$ a prime element. If $B[p^{-1}]$ is a UFD, then $B$ is a UFD.

We have: $\quad\Delta$ is prime $\quad+\quad$ $R_n[\Delta^{-1}]$ is UFD $\quad\Longrightarrow\quad$ $R_n$ is UFD.

Theorem: $R_n$ is a unique factorization domain for every $n\geq5$.

Consequences of Factoriality

$\circ$ Unique reduced denominators: for $n\geq5$, any $f\in\operatorname{Frac}(R_n)$ has a reduced representation

$$ f=\frac{N}{D}, \qquad \gcd(N,D)=1, $$

$\phantom{\circ}$ where $D$ is unique up to a nonzero constant, and its factorization $D=\prod_i\mathcal D_i^{q_i}$ is unique up to
$\phantom\circ$ constants and ordering.

$\circ$ For scattering amplitudes: fix a basis of integral / transcendental functions $\{h_i\}$,

$$ \mathcal A^{(\ell)}=\sum_i c_i h_i, \qquad c_i\in\operatorname{Frac}(R_n). $$

$\phantom{\circ}$ If the $h_i$ are linearly independent over $\operatorname{Frac}(R_n)$, each coefficient $c_i$ is uniquely defined and has
$\phantom\circ$ a unique reduced denominator.

Applies to any theory, any loop order, for massless $n\geq5$.

Constructive Factoriality

Factorization Through Principal Localization

$\circ$ More generally, suppose a quotient domain $R=S/J$ admits a principal localization

$$ R[\Delta^{-1}] \cong P[\Delta^{-1}], \qquad P=\mathbb F[y_1,\ldots,y_m]. $$

$\circ$ The localization underlying Nagata's criterion gives a constructive factorization algorithm:

0. take $f\in R$ to be factored;

1. rewrite $f$ in the localized polynomial coordinates as $f=N/\Delta^k$, with $N\in P$;

2. factor $N$ using ordinary polynomial factorization, $N=u\prod_i f_i^{m_i}$;

3. determine the valuations $v_\Delta(f_i)$ using saturation or $p$-adic evaluations in $R$,
$\kern2.3mm$ since the factors $f_i$ may contain hidden powers of $\Delta$;

4. exactly divide out these powers, combine them with $\Delta^{-k}$, and pull the remaining factors back to $R$.

$\circ$ The localization element need not itself be prime:

$\star$ if $\Delta=\prod_\alpha \Delta_\alpha^{e_\alpha}$ with each $\Delta_\alpha$ prime, we simply track the valuations $v_{\Delta_\alpha}$ separately.

$\star$ if a factor $\Delta_\alpha$ is non-prime, the same construction may still produce a valid factorization in $R$,
$\phantom\star$ but irreducibility and uniqueness are no longer guaranteed.

Finding a Localization Automatically

$\circ$ Write $R=S/J=\mathbb F[X]/J$, split the coordinates $X=Y\sqcup Z$,
$\phantom\circ$ and regard the quotient relations over the rational-function field $\mathbb F(Y)$:

$$ J \longrightarrow J^e\subset\mathbb F(Y)[Z]. $$

$\circ$ If the quotient has vector-space dimension one,

$$ \dim_{\mathbb F(Y)} \mathbb F(Y)[Z]/J^e =1, $$

$\phantom{\circ}$ then for every $z\in Z$, there exists $r_z(Y)\in\mathbb F(Y)$ such that

$$ z-r_z(Y)\in J^e, \qquad\text{i.e.}\qquad z=r_z(Y)\ \bmod J^e . $$

$\phantom{\circ}$ Thus the projection onto the $Y$-coordinates is birational.

$\circ$ Taking a common denominator $D$ of the $r_z(Y)$ gives

$$ R[D^{-1}] \cong \mathbb F[Y,D^{-1}]. $$

$\circ$ For $R_5$, choosing $Z=(c_1,c_2,d_1,d_2)$ gives

$$ D=a_1b_2-a_2b_1=\langle12\rangle . $$

Residue Rings Need Not Be Factorial

$\circ$ For a prime pole $\Delta\in R_n$, residues take values in

$$ \operatorname{Frac}(B_\Delta), \qquad B_\Delta=R_n/(\Delta). $$

$\phantom{\circ}$ Although $R_n$ is a UFD for $n\geq5$, the residue ring $B_\Delta$ need not be factorial.

$\circ$ For example, on $\langle12\rangle=0$, Schouten gives

$$ \langle13\rangle\langle24\rangle = \langle14\rangle\langle23\rangle . $$

$\phantom{\circ}$ The four brackets remain irreducible and pairwise non-associate,
$\phantom{\circ}$ so $B_{\langle12\rangle}$ has genuinely inequivalent factorizations.

$\circ$ This is not exceptional: the residue rings associated with all pole varieties encountered so far in
$\phantom{\circ}$ massless QCD amplitudes through two loops and planar six points are non-factorial, including

$\langle ij\rangle$, $\ \langle i|k+l|j]$, $\ s_{ijkl}$, $\ \Delta_{ij|kl|mn}$.

Least-Degree Denominators in Non-UFD Rings

$\circ$ Suppose the relevant factors belong to a known alphabet $\mathcal A=\{\mathcal D_1,\ldots,\mathcal D_s\}$.
$\phantom\circ$ Encode a multiplicative representation by

$$ e=(e_1,\ldots,e_s)\in\mathbb Z^s, \qquad T_e=\prod_{j=1}^s\mathcal D_j^{e_j}, $$

$\phantom{\circ}$ with positive/negative entries describing numerator/denominator factors.

$\circ$ On a univariate slice $\phi:B_\Delta \to \mathbb{F}_p[t]$, factor the alphabet elements as

$$ \phi(\mathcal D_j) = u_j\prod_{\alpha}p_\alpha(t)^{M_{\alpha j}}, \qquad u_j\in K^\times , $$

$\phantom{\circ}$ where $M_{\alpha j}$ records the multiplicity of the sliced irreducible factor $p_\alpha(t)$.

$\circ$ For a fixed matching $e_0$, all multiplicatively indistinguishable representations on that slice are

$$ e=e_0+\ker_{\mathbb Z}M . $$

$\circ$ Intersect the affine lattices from several slices, then select the admissible representation(s) with
$\phantom\circ$ minimum denominator degree. Non-uniqueness becomes a controlled integer-lattice problem.

Implementation and Examples

$\circ$ Quotient-ring factorization is implemented in syngular and lips. For

$$ \begin{aligned} f={}&-s_{12}s_{24}s_{34}-s_{12}s_{34}^2-s_{12}s_{34}s_{46} -s_{13}s_{24}s_{34}-s_{13}s_{24}s_{46}\\ &-s_{13}s_{34}^2-2s_{13}s_{34}s_{46}-s_{13}s_{46}^2 +s_{14}s_{23}s_{34}+s_{14}s_{23}s_{46}, \end{aligned} $$

$\phantom\circ\;$ R6.factorize(f, as_list=True) gives

$$ f= \bigl([1|4|6|3]-[1|5|4|3]\bigr) \bigl(\langle1|4|6|3\rangle-\langle1|5|4|3\rangle\bigr). $$

$\phantom\circ$ The factors are returned in spinor components; here they are rewritten in invariant notation.

$\circ$ In a non-UFD, the lattice algorithm implemented in antares can enumerate all least-degree
$\phantom\circ$ representations. For the four-point Parke--Taylor function:

FourPointParkeTaylor.get_lcd(Slices, ufd=False)

$$ \frac{\langle12\rangle^3} {\langle23\rangle\langle34\rangle\langle14\rangle}, \qquad \frac{[34]^3} {[12][23][14]}, \qquad \frac{\langle12\rangle^2[34]} {\langle23\rangle\langle14\rangle[12]}, \qquad \ldots $$

$\phantom\circ$ The algorithm automatically returns all eight distinct least-degree representations.

Conclusions
&
Outlook

Spinor-Helicity Hjj Remainders

$\circ$ The $pp\rightarrow Hjj$ coefficient functions are 1.5 MB of pain text LaTeX, fast and stable to evaluate.
$\phantom{\circ}$ Matrices $M_{ij}$ account for another 22 MB. Transcendental basis at PentagonFunctions++.

$\quad\small\rhd$ The size split is: 4gH ppmm 31%; pppm 22%, pppp 5%; 2q2gH pmpm 27%, pmpp 13%, 4qH pmpm 2%.

$\quad\small\rhd$ The largest (rational number) numerator (denominator) in the functions has 8 digits (6 digits);

$\quad\small\phantom{\rhd}$ while in the rational matrices they are 28 and 23 digits respectively.

$\quad\small\rhd$ Pheno ready results for the hard functions are available at FivePointAmplitudes.

A Numerical CAS for Computations in Q-Rings

(partially work in progress)

$\circ$ antares (automated numerical to analytical reconstruction software)
$\rightarrow$ Univariate slicing, LCD determination, basis change, multivariate partial fractioning strategies,
$\phantom{\rightarrow}$ constraining of numerators, Ansatz generation and fitting strategies
$\rightarrow$ Limit analytic manipulations as much as possible, mostly relies on numerical evaluations.

$\circ$ antares-results (human readable exprs in docs) with CI tests for coefficients and/or full amplitudes

In [1]: from antares_results.Hjj.remainders import remainder
In [2]: from antares_results.Hjj.momenta import oPs
In [3]: complex(remainder("ggggH", "pmpm_2L_Nc2_Nf0", oPs))
Out[3]: (-37.04012190864545-74.17026277643683j)

In [1]: import syngular
In [2]: from antares_results.Hjj.HTL.ggggH.ppmm import lTerms
In [3]: syngular.USE_ELLIPSIS_FOR_PRINT = True
In [4]: print(lTerms[:2], lTerms[-2:])

Out[4]: [Terms("""+(1⟨3|4⟩²)/(⟨1|2⟩²)"""), ('3412𝟓', True)] [Terms("""
+(-994/9⟨1|2⟩³⟨1|3⟩⟨2|4⟩[1|2]³+...⟪45 terms⟫...+45⟨1|2⟩⟨2|4⟩⟨3|4⟩³[2|4][3|4]²)/(⟨1|2⟩³⟨2|3⟩[2|3][3|4])
+(-45/2⟨1|2⟩⁴⟨1|3⟩²[1|2]⁶[1|3]+...⟪102 terms⟫...+45/2⟨1|2⟩⟨2|3⟩⟨3|4⟩⁴[1|3]²[2|3][2|4]³[3|4])/(⟨1|2⟩⟨1|3⟩[1|3]⟨2|3⟩[2|3][2|4][3|4]⟨2|1+3|2]²)
+(-172/3⟨1|2⟩³⟨1|4⟩[1|2]⁴[1|3][2|4]+...⟪37 terms⟫...+45/2⟨1|3⟩⟨3|4⟩³[1|2][1|3][2|3][3|4]³)/(⟨1|2⟩[1|3][2|4][3|4]⟨1|2+4|1]⟨1|2+4|3])
+(45/2⟨1|2⟩³⟨1|3⟩⟨1|4⟩²[1|2]³[1|3]+...⟪76 terms⟫...-1/3⟨1|2⟩⟨1|4⟩⟨3|4⟩⁴[2|3][3|4]³)/(⟨1|2⟩²⟨2|4⟩[3|4]⟨1|3+4|2]⟨1|2+4|3])
+⟨1|3⟩(-1/3⟨1|2⟩²⟨1|4⟩³[1|2]⁴+...⟪34 terms⟫...-1/3⟨1|4⟩⟨3|4⟩⁴[2|3]²[3|4]²)/(⟨1|2⟩⟨2|4⟩⟨1|3+4|2]²⟨1|2+4|3])
+(68/3⟨1|2⟩²⟨1|4⟩³[1|2]³[1|3]²[2|4]²+...⟪51 terms⟫...-2⟨1|4⟩⟨2|4⟩⟨3|4⟩³[1|2][2|3]²[2|4][3|4]³)/([1|3]⟨2|4⟩[2|4][3|4]⟨1|3+4|2]⟨1|2+4|3]²)
+(56/3⟨1|2⟩²⟨1|4⟩²[1|2]²[1|3][2|3][2|4]²+...⟪23 terms⟫...+8⟨1|3⟩⟨3|4⟩³[2|3]³[3|4]³)/([2|4][3|4]⟨1|2+4|3]³)
+(s_124-s_234)[1|2](2063/48⟨1|2⟩³⟨1|4⟩²[1|2]³[1|4]+...⟪84 terms⟫...-6293/72⟨1|3⟩⟨1|4⟩²⟨2|4⟩⟨3|4⟩[1|2][1|4][3|4]²)/(⟨1|2+4|3]²⟨2|1+3|4]Δ_13|24|𝟓)
+(s_124-s_234)(s_123-s_134)⟨1|3⟩²⟨2|4⟩[1|2][1|3](-23/16⟨1|2⟩⟨1|3⟩[1|2]²[1|3]+...⟪10 terms⟫...-69/16⟨2|4⟩⟨3|4⟩[1|2][2|4][3|4])/(⟨1|2+4|3]⟨2|1+3|4]Δ_13|24|𝟓²)
+[1|2](-253/2⟨1|2⟩³⟨1|3⟩⟨2|4⟩[1|2]⁴+...⟪16 terms⟫...-161/2⟨1|4⟩⁴⟨3|4⟩[1|4]⁴)/(⟨1|2+4|3]⟨2|1+3|4]Δ_13|24|𝟓)
+('2143𝟓', False, '+')
+('3412𝟓', True, '+')
+('4321𝟓', True, '+')
"""), ('1243𝟓', False)]

Thank you
for your attention!

These slides are powered by:
markdown, html, revealjs, hugo, mathjax, github

Backup slides.

Effective Pentagons (another non UFD example)

$\circ$ As mentioned, pentagons can be reduced to a combination of boxes,

$$ \def\mt{m} \def\mh{M_H} \def\spa#1.#2{\left\langle#1\,#2\right\rangle} \def\spb#1.#2{\left[#1\,#2\right]} \begin{eqnarray} &&E_0(p_1,p_2,p_3,p_4;\mt)= c^{(1)} D_0(p_2,p_3,p_4;\mt) +c^{(2)} D_0(p_{12},p_3,p_4;\mt) \\ &+&c^{(3)} D_0(p_1,p_{23},p_4;\mt) +c^{(4)} D_0(p_1,p_2,p_{34};\mt) +c^{(5)} D_0(p_1,p_2,p_3;\mt)\, . \end{eqnarray} $$

$\circ$ We find it useful to write the box coefficients in terms of effective pentagons $\hat e$ and boxes $\hat d$

$$ d^{h_1h_2}_{p_a\times p_b \times p_c } = \sum_{i=\{i_1,i_2\}} c^{(i)} \hat e_{p_x \times p_y \times p_z \times p_w}+ \hat d^{h_1h_2}_{p_a\times p_b \times p_c } $$

$\phantom{\circ}$ where the sum involves the two pentagons that pinch to the given box.

$\circ$ The coefficients $\hat e$ and $\hat d$ are not uniquely defined, but $\hat e$ has the property of capturing
$\phantom{\circ}$ the residue of the poles that mix top-mass and kinematic dependence.
$\phantom{\circ}$ The non-uniqueness comes from, e.g.

$$ ⟨1|2⟩[1|2]⟨1|𝟓|𝟒|𝟑|2]⟨2|𝟑|𝟒|𝟓|1]+m_t^2\text{tr}_5(1|2|𝟑|𝟒)^2=0 $$

Example of Code Syntax for Codim-2 Limit

$\circ$ This is just a couple of pip install's aways

field = Field("padic", 2 ** 31 - 1, 5)
oPs8pt = Particles(8, field=field, seed=0)
oPs8pt._singular_variety(("s_34-s_56", "s_56-s_78", '⟨1|7+8|5+6|3+4|2]', '⟨2|3+4|5+6|7+8|1]'),
                         (field.digits, field.digits, 1, 1), seed=0,
                         generators=('s_34-s_56', 's_56-s_78', '⟨1|7+8|5+6|3+4|2]', 
                                     '⟨2|3+4|5+6|7+8|1]', 'tr5(1|2|3+4|5+6)'))
oPs8pt.m_t = field.random()
oPs8pt.m_h = "sqrt(s_34)"
oPs5pt = oPs.cluster([[1, ], [2, ], [3, 4], [5, 6], [7, 8]])
from antares_results.HHH.ggHHH.pp import coeffs as coeffs_pp
coeffs_pp[’d_12x3x4’](oPsC)

130808068*2147483647^-1 + 687356881 + 792807618*2147483647 + 696603492*2147483647^2 + O(2147483647^3)

The denominator goes like $p^2$, but the coefficient goes like $p^{-1} \Rightarrow$ the numerator vanishes linearly.

$\circ$ The output is a $p$-adic number, i.e. a Laurent series in powers of the prime.
$\phantom{\circ}$ With finite fields we cannot do this (with just one evaluation)! It would be dividing by zero.

Discontinuity in the second derivative GDL, Ita, Kuschke, Ruf, Sotnikov

$\circ$ First (?) fully massless pseudo-threshold(?). Cross $\Sigma_5^{(3)}=0 \text{ at } t=0$ ,

$$ \Sigma_5^{(3)} \, = \, \big(s_{12}s_{24}+s_{13}(s_{14}+s_{24})+s_{123}(s_{12}+s_{14})\big)^2-4s_{13}s_{14}s_{123}s_{124} \, . $$

$p\kern0.5mm$-adic numbers

$\circ$ You may be familiar with finite field (integers modulo a prime)

von Manteuffel, Schabinger `14;$\;$ Peraro `16

$$ \displaystyle a \in \mathbb{F}_p : a \in \{0, \dots, p -1\} \; \text{ with } \; \{+, -, \times, \div\} $$

$\phantom{\circ}$ Limits (and calculus) are not well defined in $\mathbb{F}_p$. We can make things zero, but not small:

$$ \displaystyle |a|_0 = 0 \; \text{if} \; a = 0 \; \text{else} \; 1 \quad \text{a.k.a. the trivial absolute value.} $$

$\circ$ There exists just one more absolute value on the rationals, the $p$-adic absolute value.

Ostrowski's theorem 1916

$\circ$ Let's start from $p$-adic integers, instead of working modulo $p$, expand in powers of $p$

$$ \displaystyle a \in \mathbb{Z}_p : a_0 p^0 + a_1 p^1 + a_2 p^2 + \dots + \mathcal{O}(p^n) $$

$\phantom{\circ}$ In some sense we are correcting the finite field result with more (subleading) information.

$\circ$ $p$-adic numbers $\mathbb{Q}_p$ allow for negative powers of $p$, (would be division by zero in $\mathbb{F}_p$!)

$$ \displaystyle a \in \mathbb{Q}_p : a_{-\nu} p^{-\nu} + \dots + a_0 + a_1 p^1 + \dots + \mathcal{O}(p^n) $$

GDL, Page `22

$\circ$ The $p$-adic absolute value is defined as $|a|_p = p^\nu$.

$\phantom{\circ}$ Think of $p$ as a small quantity, $\epsilon$, (by $|\,|_p$) even if it is a large prime (by the real abs. $|\,|_\infty$).

Laurent Series or p(z)-adic expansion

$\circ\,$ With $p$-adic numbers this would be straight forward, set $\mathcal{D}_j\propto p$ and evaluate the function

$$ r_{i\in \mathcal{B}} = \sum_{m = 1}^{\text{max}_i(q_{ik})} \frac{e^k_{im}}{p^m} + \mathcal{O}(p^0) \text{ is a number in } \mathbb{Q}_p $$

$\phantom{\circ}\,$ See Particles._singular_variety or Ideal.point_on_variety to generate the configuration

$\circ\,$ We can't do this with only finite fields. Instead, build Laurent expansions around $t_{\mathcal{D}_k}$ (use more slices)

$$ r_{i \in \mathcal{B}} = \sum_{m = 1}^{\text{max}_i(q_{ik})} \frac{e^k_{im}}{(t-t_{\mathcal{D}_k})^m} + \mathcal{O}((t-t_{\mathcal{D}_k})^0) $$

$\phantom{\circ}\,$ strictly formal over $\mathbb{F}_p$, but convergent over $\mathbb{Q}_p$ for $(t-t_{\mathcal{D}_k}) \propto p$

$\circ\,$ What if the letter does not have a factor linear in $t$? E.g.

$$ r_{i \in \mathcal{B}} = \sum_{m = 1}^{\text{max}_i(q_{ik})} \frac{c^k_{im} t + d^k_{im}}{(t^2+a_kt+b_k)^m} + \mathcal{O}((t^2+a_kt+b_k)^0) $$

see also Fontana, Peraro ('23)

$\circ\,$ From these coefficients, build null spaces used in the search for simple functions

$$ \text{null}(\text{Res}(r_{i \in \mathcal{B}}, \mathcal{D}_k^m))_{ij} \text{ from } \text{ rref } (d^k_{m})_{i,\text{slice}_j} $$

mPFD from Conjectured Properties

(for planar five-point one-mass amplitudes -- properties checked a posteriori)

$\circ\,$ Denominator pairs $\{\mathcal{D}_i, \mathcal{D}_j\}$ can be cleanly separated:

$$ \frac{\mathcal{N}}{\mathcal{D}_i^{q_i}\mathcal{D}_j^{q_j}\mathcal{D}_{\text{rest}}} \rightarrow \frac{\mathcal{N}_i}{\mathcal{D}_i^{q_i}\mathcal{D}_{\text{rest}}} + \frac{\mathcal{N}_j}{\mathcal{D}_j^{q_j}\mathcal{D}_{\text{rest}}} $$

$\phantom{\circ}\,$ Examples of $\{\mathcal{D}_i, \mathcal{D}_j\}$ are:

$\qquad\star\,$ Any pairs of $s_{ijk}$ or $\Delta_{ij|kl|mn}$ or $\langle i|j|p_V|k|i]-\langle j|l|p_V|k|j]$
$\qquad\star\,$ Any conjugate pair $\{\langle i|j+k|l], \langle l|j+k|i]\}$ or cyclic $\{\langle i|j\rangle, [i|j]\}$
$\qquad\star\,$ Pairs of the form $\{\Delta_{ij|kl|mn}, \langle c|a+b|d] \text{ or } \langle ab \rangle \text{ or } [ab] \}$ unless $\{ab\}$ are $\{ij\}$ or $\{kl\}$ or $\{mn\}$

$\circ\,$ Other denominator pairs $\{\mathcal{D}_i, \mathcal{D}_j\}$ can be separated to order $\kappa$

$$ \frac{\mathcal{N}}{\mathcal{D}_i^{q_i}\mathcal{D}_j^{q_j}\mathcal{D}_{\text{rest}}} \rightarrow \sum_{\kappa - q_j\leq m \leq q_i}\frac{\mathcal{N}_i}{\mathcal{D}_i^{m}\mathcal{D}_j^{\kappa - m}\mathcal{D}_{\text{rest}}} $$

$\qquad\star\,$ E.g. $\Delta_{ij|kl|mn}^4, \langle i|k+l|j]^5$ are separable to order 5.

$Vjj$ Example

$\circ\,$ Start from the function

$$ \displaystyle f^{\text{ex}} = \frac{\mathcal{N}^{\text{ex}}}{⟨14⟩^2[14]^2 s_{56} ⟨1|2+4|3]^2⟨2|1+4|3]^4⟨2|1+3|4]^2Δ_{14|23|56}^4} $$

$\phantom{\circ}\,$ The numerator Ansatz has size 104$\,$128

$\circ\,$ Clean up the $Δ_{14|23|56}$ Gram residue

$$ \displaystyle f^{\text{ex}} = \frac{\mathcal{N}^{\text{ex}}_1}{⟨14⟩^2[14]^2s_{56}⟨2|1\!+\!4|3]^4Δ_{14|23|56}^4 \,} + \frac{\mathcal{N}^{\text{ex}}_2}{⟨14⟩^2[14]^2s_{56}⟨2|1+4|3]^4⟨1|2\!+\!4|3]^2⟨2|1\!+\!3|4]^2} $$

$\circ\,$ Split $s_{14}$ and impose symmetry

$$ \displaystyle f^{\text{ex}} = \frac{\mathcal{N}^{\text{ex}}_{3}}{⟨14⟩^2 s_{56} ⟨2|1+4|3]^4Δ_{14|23|56}^4} + \frac{\mathcal{N}^{\text{ex}}_{4}}{⟨14⟩^2 s_{56} ⟨1|2+4|3]^2⟨2|1+4|3]^4⟨2|1+3|4]^2} + (123456\rightarrow \overline{432165}) $$

$\circ\,$ Impose degree bound on poles at codimension two

$$ \displaystyle f^{\text{ex}} = \sum_{k=0}^3 \frac{\mathcal{N}^{\text{ex}}_{5,k}}{⟨14⟩^2 s_{56} ⟨2|1+4|3]^{1+k} Δ_{14|23|56}^{4-k}} + \frac{\mathcal{N}^{\text{ex}}_6}{⟨14⟩^2 s_{56}⟨1|2+4|3]^2⟨2|1+4|3]^4⟨2|1+3|4]^2} + (123456\rightarrow \overline{432165}) $$

The Ansatz now has size 13$\,$532, almost a factor of 10 simpler.

The Numerator Ansatz

$\circ\,$ The numerator Ansatz takes the form (massless case)

GDL, Maître ('19)

$\displaystyle \text{Num. poly} = \sum_{\vec \alpha, \vec \beta} c_{(\vec\alpha,\vec\beta)} \prod_{j=1}^n\prod_{i=1}^{j-1} \langle ij\rangle^{\alpha_{ij}} [ij]^{\beta_{ij}}$

$\phantom{\circ}$ subject to constraints on $\vec\alpha,\vec\beta$ due to: 1) mass dimension; 2) little group; 3) linear independence.

$\circ\,$ Construct the Ansatz via the algorithm from Section 2.2 of GDL, Page ('22)

Linear independence = irreducibility by the Gröbner basis of a specific ideal.

$\circ\,$ Efficient implementation using open-source software only

Gröbner bases $\rightarrow$ constrain $\vec\alpha,\vec\beta$
Decker, Greuel, Pfister, Schönemann

Integer programming $\rightarrow$ enumerate sols. $\vec\alpha,\vec\beta$
Perron and Furnon (Google optimization team)

$\circ\,$ Linear systems solved w/ CUDA over $\mathbb{F}_{2^{31}-1}$ ($t_{\text{solving}} \ll t_{\text{sampling}}$) w/ linac (coming soon)

Preview of Linac

work in collaboration with Jack Franklin, avaiable now! @ github.com/GDeLaurentis/linac

cuda_row_reduce(A, field_characteristic=primes[0], verbose=False)  # A is a 2D numpy.ndarray

$\circ\,$ Performance on a laptop GPU of approx. 60 CPU cores
$\circ\,$ Performance on a workstation GPU of approx. 600 CPU cores
$\circ\,$ Tested on systems up to 100k equations and unknowns (takes 45 minutes).

Iterated Pole Subtraction

(i.e. geometry at codimension greater than one)

GDL, Maître ('19) GDL, Page ('22) Chawdhry ('23) Xia, Yang ('25)

$\circ\,$ Iteratively reconstruct a residues at a time using $p$-adic numbers to get $\mathbb{F}_p$ samples for the residues

$$ \begin{alignedat}{2} & r^{(139 \text{ of } 139)}_{\bar{u}^+g^+g^-d^-(V\rightarrow \ell^+ \ell^-)} = & \qquad\qquad & {\small \text{Variety (scheme?) to isolate term(s)}} \\[2mm] & +\frac{7/4{\color{blue}(s_{24}-s_{13})}⟨6|1+4|5]s_{123}{\color{green}(s_{124}-s_{134})}}{⟨1|2+3|4]⟨2|1+4|3]^2 Δ_{14|23|56}} + & \qquad\qquad & \Big\langle ⟨2|1+4|3]^2, Δ_{14|23|56} \Big\rangle \\[1mm] & -\frac{49/64⟨3|1+4|2]⟨6|1+4|5]s_{123}(s_{123}-s_{234})(s_{124}-s_{134})}{⟨1|2+3|4]⟨2|1+4|3]Δ^2_{14|23|56}} + \dots & \qquad\qquad & \Big\langle Δ_{14|23|56} \Big\rangle \end{alignedat} $$

$\circ\,$ We get more than just partial fraction decomposition, we can identify numerator insertions from e.g.:

$$ \sqrt{\big\langle ⟨2|1+4|3], Δ_{14|23|56} \big\rangle} = \big\langle {\color{green}(s_{124}-s_{134})}, ⟨2|1+4|3] \big\rangle \, , \\[1mm] \big\langle ⟨1|2+3|4], ⟨2|1+4|3] \big\rangle = \big\langle ⟨1|2+3|4], ⟨2|1+4|3], {\color{blue}(s_{13}-s_{24})}\big\rangle \cap \big\langle ⟨12⟩, [34] \big\rangle $$

$\circ\,$ Interesting and non-trivial bevhavior also at 5-point 3-mass

$$ \def\spa#1.#2{\left\langle#1\,#2\right\rangle} \def\spb#1.#2{\left[#1\,#2\right]} \def\spaa#1.#2.#3{\langle\mskip-1mu{#1} | #2 | {#3}\mskip-1mu\rangle} \def\spbb#1.#2.#3{[\mskip-1mu{#1} | #2 | {#3}\mskip-1mu]} \def\spab#1.#2.#3{\langle\mskip-1mu{#1} | #2 | {#3}\mskip-1mu]} \def\spba#1.#2.#3{[\mskip-1mu{#1} | #2 | {#3}\mskip-1mu\rangle} \def\spaba#1.#2.#3.#4{\langle\mskip-1mu{#1} | #2 | #3 | {#4}\mskip-1mu\rangle} \def\spbab#1.#2.#3.#4{[\mskip-1mu{#1} | #2 | #3 | {#4}\mskip-1mu]} \def\spabab#1.#2.#3.#4.#5{\langle\mskip-1mu{#1} | #2 | #3 | {#4}| {#5} \mskip-1mu]} \def\spbaba#1.#2.#3.#4.#5{[\mskip-1mu{#1} | #2 | #3 | {#4}| {#5}\mskip-1mu\rangle} \def\tr#1.#2{\text{tr}(#1|#2)} \def\qb{\bar{q}} \def\Qb{\bar{Q}} \def\cA{{\cal A}} \def\slsh{\rlap{$\;\!\!\not$}} \def\three{{\bf 3}} \def\four{{\bf 4}} \def\five{{\bf 5}} \begin{align}\label{eq:decomp_spaba1351_spbab2542} \big\langle \spaba1.\three.\five.1,\, \spbab2.\five.\four.2 \big\rangle = \; &\big\langle \, \spab1.\three.2,\, \spab1.\four.2,\, \spaba1.\three.\five.1,\, \spbab2.\five.\four.2 \, \big\rangle\; \cap \\ &\big\langle \, \spaba1.\three.\five.1,\, \spbab2.\five.\four.2, |\five|2]\langle1|\three| - |1+\three|2]\langle1|\five| \, \big\rangle \;, \nonumber \end{align} \\ \text{because: } |\five|2]\spaba1.\three.\five.1[2| + |1\rangle\spbab2.\five.\four.2\langle1|\five| = \spab1.\five.2 \Big( |\five|2]\langle1|\three| - |1+\three|2]\langle1|\five| \Big) \, , $$

$\phantom{\circ}\,$ or between the triangle and box Grams

$$ \begin{gather}\label{eq:decomp_delta12_34_5_and_delta_12_3_4_5} \big\langle \Delta_{12|34|5},\,\Delta_{12|3|4|5} \big\rangle = \big\langle s_{34},\, \tr1+2.{\three+\four}^2 \big\rangle \cap \big\langle \Delta_{12|34|5},\, \tr1+2.{\three-\four}^2 \big\rangle \, . \end{gather} $$

Iterated Pole Subtraction (another example)

$\circ$ Example from triple-Higgs

$$ \hat d^{++}_{12\times 3 \times 4}= \frac{\mathcal{N} \leftarrow 2794 \text{ free parameters }}{⟨12⟩²⟨1|𝟓|𝟒|𝟑|2]⟨2|𝟑|𝟒|𝟓|1]Δ_{12|𝟑|𝟒|𝟓}} $$

$\circ$ We can prove $⟨1|𝟓|𝟒|𝟑|2], ⟨2|𝟑|𝟒|𝟓|1]$ can be separated, their primary decomposition reads

$$ \big\langle ⟨1|𝟓|𝟒|𝟑|2], ⟨2|𝟑|𝟒|𝟓|1] \big\rangle = \big\langle ⟨1|𝟓|𝟒|𝟑|2], ⟨2|𝟑|𝟒|𝟓|1], \text{tr}_5 \big\rangle \cap \big\langle ⟨1|𝟓|𝟒|𝟑|2], ⟨2|𝟑|𝟒|𝟓|1], s_{2𝟑}, s_{1𝟓} \big\rangle $$

$\phantom{\circ}$ Generate two phase space points, one for each branch, and verify the numerator vanishes.

$\circ$ Similarly, with four evaluations we can prove $⟨1|𝟓|𝟒|𝟑|2], Δ_{12|𝟑|𝟒|𝟓}$ can be separated,

$$ \big\langle ⟨1|𝟓|𝟒|𝟑|2] , \, Δ_{12|𝟑|𝟒|𝟓} \big\rangle= \big\langle M_H, \; 𝟓_{\alpha\dot\alpha}𝟒^{\dot\alpha\beta} \big\rangle \cap \big\langle M_H, \; 𝟒^{\dot\alpha\alpha}𝟑_{\alpha\dot\beta} \big\rangle \cap \big\langle \langle 1 | 𝟑 | 2], \; \langle 1 | 𝟒 | 2], \; \langle 1 | 𝟑 | 𝟒 | 1 \rangle, [2 | 𝟑 | 𝟒 | 2] \big\rangle \cap \big\langle ??? \big\rangle $$

$\phantom{\circ}$ Although we don't have a complete set of generators for the last branch, we can still sample it.

$\circ$ Fit $⟨1|𝟓|𝟒|𝟑|2]$ residue by sampling in limit $⟨1|𝟓|𝟒|𝟑|2] \rightarrow 0$

$$ \hat d^{++}_{12\times 3 \times 4} = \frac{\mathcal{N} \leftarrow 112 \text{ free parameters }}{⟨12⟩²⟨1|𝟓|𝟒|𝟑|2]} + \mathcal{O}(⟨1|𝟓|𝟒|𝟑|2]^0) $$

Core Tools - Fully Open Source

For fleshed out examples see e.g. GDL (ACAT '22) or Appendix B of 2504.19909

Install from github (git clone) or PyPI (pip install); use of Jupyter is recommended.

$\circ$ pyadic
$\quad\rightarrow$ Finite field $\mathbb{F}_p$ and $p$-adic $\mathbb{Q}_p$ number types, including field extensions
$\quad\rightarrow$ rational number reconstruction (Wang's EEA, LGRR, MQRR)
$\quad\rightarrow$ univariate and multivariante Newthon & univariate Thiele interpolation algorithms in $\mathbb{F}_p$

$\circ$ syngular (in the backhand Singular is used for many operations)
$\quad\rightarrow$ object-oriented algebraic geometry (Field, Ring, Quotient Ring, Ideal)
$\quad\rightarrow$ ring-agnostic monomials and polynomials (with support for unicode characters, e.g. spinor brackets)
$\quad\rightarrow$ multivariate solver (Ideal.point_on_variety), under- and over-constrained systems OK
$\quad\rightarrow$ a semi-numerical prime and primary ideal test (assumes equi-dimensionality of ideal)

$\circ$ lips (Lorentz invariant phase space)
$\quad\rightarrow$ phase space points over any field ($\mathbb{Q}, \mathbb{Q}[i], \mathbb{R}, \mathbb{C}, \mathbb{Q}_p, \mathbb{F}_p$), including internal and external masses
$\quad\rightarrow$ evaluate any Mandelstam or spinor expression (custom ast/regex parser)
$\quad\rightarrow$ generation of any special kinematic configuration (wrapper around Ideal.point_on_variety)

What can algebraic geometry do for rational functions in scattering amplitudes?

Part 1: Computing Amplitudes