How many nonzero points are there on $$x^3 y + y^...

Counting solutions for the equation x^3 y + y^3 z + z^3 x = 0 in GF(5^18). Considering projective space, we're counting nonzero solutions up to scaling by the field size q = 5^18.

Considering affine solutions first, then mapping to projective geometry. Now evaluating substitutions and factorization to streamline the process of counting unique solutions.

Analyzing polynomial factorization

I’m noting that the homogeneous polynomial x^3 y + y^3 z + z^3 x of degree 4 in projective space over GF(5^18) does not factorize in a simplifiable manner.

Mapping polynomial structures

I’m considering the polynomial's structure and testing points in the projective plane to identify solutions. This involves examining homogeneous polynomials and the polynomial's symmetry.

Analyzing curve characteristics

I’m examining a homogeneous polynomial of degree 4 in 3 variables, focusing on its genus and singularities, and applying the Hasse-Weil bound to determine the number of rational points in projective space.

Identifying singularities

I’m mapping out the polynomial to see any obvious singularity. This helps refine the genus, which impacts the Hasse-Weil bounds, leading to exact counts of rational points in projective space.

Verifying points

I’m checking if standard projective points (0:0:1), (0:1:0), and (1:0:0) satisfy the polynomial, establishing that these points lie on the curve.

Assessing non-singularity

I’m evaluating the partial derivatives at standard projective points (1:0:0), (0:1:0), and (0:0:1), confirming these points are not singular by ensuring at least one partial derivative is non-zero at each point.

Analyzing singularities

I’m piecing together a solution for $x = y = z$ to check for generic singular points. In GF(5^18), 3 is non-zero, so the trivial solution $x = 0$ is the only viable option.

Handling variable conditions

Now I’m working through cases where one variable might be zero, focusing on avoiding the trivial solution and ensuring non-zero values for $(x, y, z)$.

Laying out the options

OK, let's ensure $x, y,$ and $z$ are all non-zero. From there, it's interesting to see that results show $z^3 = -3 x^2 y$, $x^3 = -3 y^2 z$, and $y^3 = -3 z^2 x$.

Testing variable relationships

I’m piecing together ideals to express each variable in terms of the others and see how these relationships influence their values. This helps in simplifying the equation.

Taking a closer look

Hmm, I’m considering integrating the equations and simplifying. By multiplying coefficients and terms, I’m working through contradictions or confirming solutions if none emerge.

Evaluating field properties

I'm considering the implications of GF(5^18), treating -3 as 2 mod 5. This suggests (-3)^3 equals 1 in this setting, which is interesting.

Navigating contradictions

It's worth noting that in GF(5^18), 3 doesn't equate to 1, leading to a contradiction when assuming all variables are non-zero. This confirms there are no singular points.

Assessing the curve

I’m identifying the curve as a nonsingular plane quartic with genus 3. Using the Hasse-Weil theorem, I determine the range of GF(5^18)-rational points.

Determining the count

I’m thinking through the projective solutions of the polynomial equation over GF(5^18) and aim to pinpoint the number of nonzero points up to scaling.

Examining symmetrical insights

I’m working through the polynomial's symmetry, considering group structures, and exploring genus 3 curves. This involves checking for Frobenius cover, hyperellipticity, or trigonal properties.

Mapping out curve properties

I'm curious if the polynomial is a triple cover of P^1 or an elliptic curve. Given its genus 3, I’m leaning towards those possibilities.

Tracing curve characteristics

I’m mapping out the curve's properties, including possible singularities and its genus. The focus is on determining the number of nonzero points in the projective space over GF(5^18).

Spotting unique points

I’m considering if the curve might have special points or fit a known result like Frobenius on the Jacobian, but the axis points suggest otherwise.

Transforming variables

I’m thinking about simplifying $x^3 y + y^3 z + z^3 x$ with variable changes. The challenge is finding a known curve type in the classification of plane quartics.

Evaluating solutions

I’m looking at the equation with $x = 1$ and testing values for $z$ in GF(5^18) to see if $z^3 + z + 1 = 0$ has any solutions.

Breaking down the problem

Now I'm thinking about the Hasse-Weil bound and its limitations. The curve could be supersingular or have an L-polynomial with known forms, but we'll need to apply the bound to narrow down solutions.

Wondering about superspecial curves

I’m mapping out if the curve might be superspecial, leading to 1 + q points. Testing with $q = 5$ and directly counting projective solutions reveals some interesting insights.

Tracing projective patterns

I’m investigating distinct projective points on the curve by scaling variables $x, y,$ and $z$ in GF(5). This approach aims to reveal any emerging patterns.

Calculating solutions

I’m examining valid projective points in GF(5) by setting $x = 1$ and varying $y$ and $z$. The goal is to identify pairs $(y, z)$ that satisfy the polynomial equation.

Evaluating pair outcomes

I’m examining the polynomial equation $E(y,z)$ for $y = 0$ across $z$ values in GF(5), identifying pairs that meet the condition.

Identifying valid pairs

I’m evaluating $E(y,z)$ for $y = 1, 2, 3,$ and $4$ with different $z$ values in GF(5). This helps pinpoint pairs that satisfy the equation.

Taking a closer look

I’m mapping out valid projective points for $x = 0$ and $x \neq 0$, ensuring each scenario is distinct and accurate.

Ensuring completeness

I’m verifying all possible projective points on the curve by methodically considering each case. This thorough approach helps confirm every distinct solution is included and accurately identified.

Confirming solutions

I’m verifying valid projective points by examining symmetric cases and ensuring no solutions are overlooked. Each scenario is carefully considered for completeness and accuracy.

Piecing together distinct solutions

I’m confirming that unique projective points emerge for each scenario, with no overlaps. For $x = 1$, four solutions are identified, and for symmetry, two solutions are verified.

Summing up solutions

Combining valid solutions from different scenarios, I found 6 unique projective points over GF(5), which aligns well with the Hasse-Weil bound for genus 3.

Analyzing Frobenius roots

I'm thinking about the Frobenius roots for GF(5^18) and their correlation with genus and solutions. This requires advanced techniques like the Hasse-Weil theorem to pinpoint exact counts.

Examining polynomial attributes

Mapping out the polynomial's cyclic group antecedents reveals intriguing connections to plane curve classifications and group actions, aiding in the journey to find the zeta function.

Weighing possibilities

Considering if the curve might be supersingular with known results or if there's a minimal curve in the extension, which could influence the nonzero projective points in GF(5^18).

Reaching conclusions

Interestingly enough, the trace of Frobenius is 0 over GF(5), hinting at a rich structure. This leads me to conclude that the number of nonzero projective points in GF(5^18) is likely 6.

Analyzing Frobenius eigenvalues

I’m looking into the trace of Frobenius being 0, which impacts the number of nonzero projective points in GF(5^18) and suggests the presence of 6 distinct complex roots.

Taking a closer look

Considering $\alpha_i$ as a root of unity times $\sqrt{5}$, I’m assessing their symmetrical arrangement in the complex plane for the sum to vanish.

Taking a closer look

I'm examining constraints on $\alpha_i$ and their symmetrical arrangement in the complex plane. This involves ensuring the product $\alpha_1 \alpha_2 \ldots \alpha_6 = 125$.

Mapping polynomial structure

I’m detailing the zeta function $Z(C, T)$ for genus $g$, noticing the polynomial $P(T)$ and its constant term, which helps in understanding the product of the roots $\alpha_i$.

Piecing together the puzzle

Locking in on the polynomial $\alpha_1 \alpha_2 \ldots \alpha_6 = p^3$ and their symmetry in the complex plane, each with $\alpha_i$ as roots of unity times $\sqrt{5}$.

Evaluating projective points

I'm mapping out direct enumeration for GF(5^2), considering the distribution of roots of unity, and aiming to simplify calculations by isolating the dominant term.

Evaluating curve patterns

I’m analyzing the curve's structure, especially for elliptic and genus 3 forms, using roots of unity in GF(5^18) to identify any repeating patterns that simplify calculations.

Wondering about roots of unity

Focusing on roots of unity in GF(5^18). Mapping out patterns and their sums, I aim to pinpoint any repeating themes or trends in the distribution of nonzero points.

Evaluating patterns

I’m mapping out the exponents of $\alpha$ to identify repeating trends. I’m thinking $\alpha^{18}$ simplifies with factors of $\pi$ that are either $-1$ or $+1$ if $k$ is even or odd.

Reaching conclusions

OK, let me see. I'm piecing together that the sum of 6 equally spaced vectors around a circle is 0. For $\alpha_i$, thinking they might be $\sqrt{5} e^{i(\pi(2k+1))/6}$.

Analyzing patterns

I'm analyzing that the sum of 6 uniquely spaced vectors in the complex plane might be 0. Varied distances and angles likely contribute to this symmetry.

Wondering about angles

Initially thinking about equidistant vectors summing to 0. This leads to insights on uniquely spaced angles and their symmetrical properties in the complex plane.

Evaluating sums

I’m figuring out sums of exponents and their symmetry. Mapping out how roots of unity and specific spacing values impact the total. It’s all about real numbers and complex plane angles.

Piecing together symmetry

I’m mapping out symmetrical arrangements of vectors on a circle and their effects on polynomial roots. This zero sum hints at unique properties tied to the polynomial's coefficients.

Assessing symmetrical patterns

OK, let me see—I'm analyzing symmetrical patterns of vectors on a circle, specifically how symmetrically distributed angles can impact polynomial roots and their coefficients in the real and complex planes.

Breaking down calculations

I'm simplifying 5^18 into smaller parts for accuracy, ensuring each multiplication step is carefully verified to avoid mistakes. This method is working well so far.

Wrapping up calculations

I’m working through the final steps of 5^18 - 11,718,750 + 1, validating each multiplication and sum. This approach ensures precision and avoids errors.

Breaking down the equation

I’m mapping out the formula for alpha_j^n and confirming that the sum equals 5^n + 1 - 6 * 5^(n/2), especially when n is a multiple of 6.

Adjusting my approach

Noticing that the magnitude of alpha_j as sqrt{5} doesn't yield an integer for n=1. Considering conjugate pairs whose product is 5, leading to a sum of zero for specific n values.

Piecing together the symmetry

I’m working through how the sum of 6 complex numbers with magnitude sqrt{5} forms a symmetrical zero vector. This leads to insights about the symmetric arrangement of angles and their effects.

Confirming complex sums

I’m verifying if the sum of 6 complex numbers with magnitudes of sqrt{5} and varying angles results in an integer, considering their conjugate pairs and integer multiples of 2\pi.

Analyzing curve properties

I’m digging into the number of points on the curve over extended fields, considering symmetrical angle distribution and minimal polynomial constraints to determine if the sum of complex exponentials equals zero.

Piecing together polynomial insights

I’m gathering info on the trace of Frobenius for genus 3 and using Newton identities to recursively find sums of powers of roots. This leads me to explore characteristic polynomial coefficients.

Piecing together the puzzle

Noting that the product of six roots equals p^3, I explore simpler polynomial expressions, particularly focusing on the standard "Fermat" approach for computing solutions.

Crunching the numbers

I’m working through the polynomial N= 5^18 + 1 - 6(5^9) to simplify the expression while confirming it matches the expected results for n=1.

Thought for 11m 10s

Assessing the equation

I’m gathering info on the equation $x^3 y + y^3 z + z^3 x = 0$ in $\mathbb{F}_{5^{18}}^3$, focusing on nonzero points to ensure they align with projective geometry standards.

Investigating curve irregularities

I'm thinking through the possibility of singular solutions or infinite points on the curve $x^3 y + y^3 z + z^3 x = 0$. Calculating partial derivatives helps to clarify these aspects.

Assessing zero points

OK, let me see. I’m checking if $x^3 y + y^3 z + z^3 x = 0$ and its partial derivatives have common zeroes. This helps figure out if the curve has singular points or infinite points.

Mapping factorization

I'm thinking through factorization in characteristic 5, testing simpler values like $y = 1$ and $x = 1$ to spot potential factorizations. Progressing steadily, let's see where this leads.

Investigating irreducibility

I'm confirming that the curve $x^3 y + y^3 z + z^3 x = 0$ is likely an irreducible plane curve of degree 4, suggesting it has genus 3 if smooth.

Pinpointing solutions

I'm identifying nontrivial solutions where any two of $x, y, z$ are zero, leading to the remaining variable being nonzero. This yields distinct points of interest for the curve.

Verifying singularity

OK, let me see. I'm checking partial derivatives for points like $(0,1,0)$. If derivatives aren't zero vectors, the point isn't singular. This ensures I'm pinpointing nonzero outcomes.

Ensuring non-singularity

I’m confirming partial derivatives are non-zero vectors for points like $(0,0,1)$, eliminating any singular points and verifying a discrepancy in the general method used for the solution.

Sysadmin tasks

I’m setting up a function to check the Frobenius characteristic polynomial for genus 3 curves. Systematically evaluating solutions over $\mathbb{F}_5$ using a table for $y$ and $z$ values.

Evaluating solutions

I'm working through $f(0,z)$ by testing $y$ and $z$ values in $\{0,1,2,3,4\}$. If $y = 0$, then $z^3 = 0$ leads to the solution $(y,z) = (0,0)$.

Assessing solutions

I'm checking $f(1,z) = 1 + z + z^3$ for $z$ in $\{0,1,2,3,4\}$. None of the equations yield $0$. Let's see what other insights emerge.

Testing values

I’m pulling together insights on $f(2,z) = 2 + 3z + z^3$ with $z$ values $\{0,1,2,3,4\}$. None of the outcomes result in zero.

Identifying solutions

Testing $f(3,z) = 3 + 2z + z^3$ with $z$ values $\{0,1,2,3,4\}$ shows that $y=3$ offers two solutions: $(3,2)$ and $(3,4)$.

Assessing outcomes

OK, let me see. Testing $f(4,z) = 4 + 4z + z^3$ with $z$ values $\{0,1,2,3,4\}$ reveals one solution: $(4,2)$. Summing up, for $x = 1$, I identified four distinct solutions for $(y,z)$.

Mapping out solutions

I’m working on solutions for $x = 0$, simplifying the polynomial to $y^3 z = 0$. This results in $(0,1,0)$, ensuring a comprehensive approach.

Piecing together

Sorting out solutions for $x = 0$, I simplify to $y^3 z = 0$. This yields $(0,1,0)$, marking progress in understanding zero constraints for projective points.

Identifying solutions

Mapping out solutions for each $x$ value in $\{0,1,2,3,4\}$, I'm testing all $y$ pairs to ensure comprehensive identification of valid solutions.

Mapping solutions

I’m verifying the distinct solutions from each chart and checking for any overlaps or double-counts by examining their scaling relationships in the projective space.

Confirming distinct points

I’ve verified that the solutions from chart $x = 1$ are unique and not overlapping with the chart $y = 1$. This ensures all valid solutions are accounted for accurately.

Confirming data accuracy

I’ve verified the distinctions for chart $z = 1$ and checked for overlaps with earlier charts. This solidifies the count of unique solutions, ensuring no data points are repeated.

Ensuring unique solutions

OK, let's make sure no solutions repeat by carefully evaluating sub-projective spaces and addressing potential overlaps. This leads to accurate and distinct results.

Pulling together solutions

I’m verifying everything to confirm unique solutions. For $x \neq 0$, I found four distinct points. For $x = 0$, there are two unique points: $(0,0,1)$ and $(0,1,0)$.

Identifying unique solutions

I’m piecing together six distinct solutions: four from $x \neq 0$ and two from $x = 0$. This confirms the solutions align with the expected results in $\mathbb{F}_5$.

Confirming validity

I’m systematically verifying that the solution $(1,3,2)$ meets the equation $x^3 y + y^3 z + z^3 x = 0$ by substituting and simplifying each expression to ensure it satisfies the condition in $\mathbb{F}_5$.

Validating the outcomes

OK, let me see. I’m checking solutions $(1,3,4)$ and $(1,4,2)$ by plugging in and simplifying, ensuring each expression aligns with the equation in $\mathbb{F}_5$. This confirms six unique solutions.

Evaluating assumptions

I'm testing if the curve's Frobenius polynomial could differ from the expected 6 solutions over $\mathbb{F}_5$. This involves examining assumptions about roots and their properties.

Piecing together the L-polynomial

Considering the L-polynomial for genus 3 curves, I'm examining the implications of roots with absolute values $\sqrt{5}$ and how the trace coefficient $T$ affects the polynomial.

Examining polynomial degrees

I’m identifying the polynomial's degree and constant term, noting that for genus 3, the product of the roots is $5^3$. This is key to finding solutions over $\mathbb{F}_{5^{18}}$.

Evaluating assumptions

Removing constraints on roots suggests checking smaller values first. Simplifying the equation will streamline finding the diamond's area.

Analyzing polynomial structures

I’m digging into polynomial structures to ensure the sum of roots equals zero. This involves managing conjugate pairs and their angles, leading to a more precise form.

Clarifying the calculation

I’m noting the need to identify the number of points in $\mathbb{F}_{5^{18}}$ by possibly examining transformations of the polynomial and testing specific values to ensure the sum of roots equals zero.

Piecing together insights

I’m thinking about how equally spaced 6th roots of unity times $\sqrt{5}$ might sum to zero for certain $n$ values when $n$ is a multiple of 6.

Breaking down possibilities

Given that $6 \divides n$ results in a manageable sum, I’m considering if the roots can be expressed symmetrically and the implications for summing these roots over various $n$ values.

Mapping out possibilities

I’m piecing together the curve's polynomial structure, factoring considerations, and irreducibility to estimate the number of points in $\mathbb{F}_{5^{18}}$.

Evaluating assumptions

Hmm, it seems the assumption about the sum of roots might not hold true for certain extensions. This points to a need for a more refined formula for the number of points in $\mathbb{F}_{5^n}$.

Mapping symmetrical arrangements

I'm piecing together the optimal symmetrical layout for the points. Distributing them in a specific pattern ensures the sum of their $n$-th powers equals zero for all $n$.

Evaluating symmetrical setups

I'm thinking through symmetrical distributions of points and their implications. Testing specific angular positions to ensure the sum of their powers equals zero. This leads me to refine the approach.

Piecing this together

Testing smaller values like $\#C(F_{25})$ to identify the sum of specific powers of roots. This should help in figuring out the total number of points in $\mathbb{F}_{5^{18}}$.

Crunching the numbers

I’m working through the Hasse-Weil bound for genus 3 curves, mapping out the potential range for $\#C(F_{5^{18}})$. The assistant's formula suggests the minimal value in this range.

Evaluating polynomial strategies

OK, let's consider the minimal polynomial of Frobenius and how symmetrical distributions of roots might tweak the count of points in $\mathbb{F}_{5^{18}}$.

Pondering the gap

I'm thinking about the noticeable difference in the results and calculating the range for $\#C(F_{5^{18}})$ using established formulas and constraints. This leads me to a more precise conclusion.

Pinpointing coefficients

I'm mapping the polynomial's coefficients with the Hasse-Weil bound, factoring into quadratic terms, and focusing on plane curves' dimension of holomorphic differentials to pinpoint the precise number of points.

Connecting the dots

I’m linking the sum and product of the polynomial's roots to figure out the constant term and coefficient of $T^5$. It’s interesting to see how these calculations align.

Assessing polynomial coefficients

I'm pulling together the sum and product of roots to determine polynomial coefficients, aiming to pinpoint the number of points for the genus 3 curve. This leads to a clearer understanding.

Understanding supersingularity

Yes, I’m figuring out how supersingularity for genus 3 curves in characteristic 5 could link to summed polynomial roots, especially the traces of Frobenius.

Mapping supersingular curves

I’m exploring the link between supersingular curves in characteristic 5, polynomial roots, and the Hasse-Weil bound to estimate the curve's points accurately. Progress is steady and promising.

Evaluating polynomial insights

Piecing together polynomial forms like the Fermat quotient to estimate points, while ensuring sums of powers of roots equal zero for genus 3 curves in characteristic 5.

Confirming the formula

It's worth noting that the discrepancy in the final result is minor, suggesting the correct formula likely involves $5^{18} + 1$ or $5^{18} + 1 + 6 \times 5^9$.

Aligning polynomial structure

Mapping out the polynomial's structure for genus 3 curves, ensuring the sum of roots equals zero. This confirms the minimal number of points is $5^n + 1$ for even $n$.

Evaluating theoretical approaches

It's interesting to see that there's a conceptual clash with the L-polynomial approach. Confirming that the number of solutions should be an integer, and the sum of $\alpha_i^2$ should be within a bounded range.

Assessing polynomial patterns

I'm analyzing the polynomial for genus 3 curves, calculating points in $\mathbb{F}_{5^{18}}$, and assessing the influence of summed root powers using established formulas and constraints.

Reflecting on supersingularity

I'm curious about supersingular curves for genus 3. They might have exactly $5^n + 1$ points for all $n$, with the Jacobian as all $p$-torsion.

Tracking the pattern

OK, let's think this through: the number of points on an elliptic curve with a trace of zero changes by $2p \cos(2\theta)$.

Verifying supersingularity

Let's see: for genus 3 curves in characteristic 5, the number of points isn't consistently $5^n + 1$ due to variations in polynomial structures and coefficients.

Hypothesizing the formula

Assuming the number of points on a genus 3 curve is $5^{18} + 1$ with a small adjustment, leading to a minor difference from the assistant's formula.

Crunching the numbers

Calculating the number of points on a genus 3 curve, I find $\#C(\mathbb{F}_{5^{18}}) = 5^{18} + 1 + 6 \cdot 5^9 = 3,814,708,984,376$, ensuring it fits the Hasse-Weil bound.

Thought for 11m 39s