02 / Maths Lab · ECO & Complexity
Bound it. Count it.
See the geometry.
Explore extended coarse order functions from my work on intersections of linear series.
The exact definition of an ECO function
Let F be a field or Z, and let R be a commutative unital F-algebra. An ECO function is a map
such that there is a constant K ≥ 0 satisfying the three conditions alongside. Conditions 2 and 3 concern x, y with finite γ.
Its bounded sublevels are γ≤n = {x ∈ R : γ(x) ≤ n}. In general these are sets; the first example below gives vector spaces.
- Finite on scalarsγ(F) ⊂ [1, ∞)Each scalar has finite value; no uniform scalar bound is imposed here.
- Coarse additionγ(x + y) ≤ max{γ(x), γ(y)} + K
- Subadditive multiplicationγ(xy) ≤ γ(x) + γ(y)
R = k[x, y] · K = 0
Choose how degree is measured.
Give x and y positive weights wx and wy. For a nonzero polynomial P = ∑ cijxiyj, its weighted degree is
γw(P) = max{1, degw P}, γw(0) = 1.
Each choice of positive weights gives an ECO function with addition defect K = 0. For integer n ≥ 1, its bounded sublevel is
The controls use positive integer weights from 1 to 6. The pair (1, 1) recovers ordinary total degree.
Current monomial condition: i + j ≤ 3.
Each point is a basis monomial. Its weighted degree is wxi + wyj. All linear combinations of the displayed monomials belong to (γw)≤n. The green region is the continuous triangle Tn; the dashed outline is the lattice polygon Πn. The view uses the same unit scale on both axes.
Monomial names are shown beside the points.
Select a monomial to inspect its weighted degree.
Convex geometry ↔ algebra ↔ intersections
One triangle. Three measurements.
Put Vn = (γw)≤n and Tn = {(u, v) ≥ 0 : wxu + wyv ≤ n}. A polynomial lies “in the triangle” when every exponent in its support belongs to Tn. For the intersection maximum, work over an infinite field k, such as Q or C.
| Measurement | Exact value | Divided by n² |
|---|---|---|
| Area AnArea of the green triangle Tn | 9/2 | 1/2≈ 0.500000 |
| Lattice points Nn#(Tn ∩ Z²) = dimk Vn | 10 | 10/9≈ 1.111111 |
| Largest finite quotient Mnsup dimk k[x, y]/(P, Q), for P, Q ∈ Vn | 9 | 1≈ 1.000000 |
The lattice polygon has area 9/2, so the exact intersection maximum is M₃ = 9.
The Hilbert–Samuel connection
The point count is the dimension of an ECO sublevel. In this example the ECO Hilbert–Samuel theorem identifies its leading growth coefficient with the self-intersection degree, with the dimension factorial 2!:
= 2 area(Δw) = 1/(wxwy).
For weights (1, 1): Aₙ/n² = 1/2, Nₙ/n² → 1/2, and Mₙ/n² → 1.
The area and normalized point count have the same limit. The normalized intersection maximum tends to twice that limit. Thus the convex body, growth of bounded polynomial spaces, and intersection degree determine one another asymptotically.
Why the finite maximum uses the lattice polygon
Let Πn = conv(Tn ∩ Z²). The exact finite-bound formula is
= 2 area(Πn) ≤ 2 area(Tn) = n²/(wxwy).
This application of the Bernstein–Kushnirenko theorem counts intersection multiplicities. The allowed supports are closed under decreasing exponents, so translating x and y preserves Vn. Any finite set of common zeros can therefore be moved away from the coordinate axes, where the Newton-polygon bound applies. Generic pairs attain it; an infinite coefficient field guarantees that such pairs exist.
A common curve component makes the quotient infinite-dimensional and is excluded. The unit ideal is allowed and has quotient dimension 0. If n < max(wx, wy), only one variable, or no variable, occurs: every finite quotient is zero, so Mn = 0. Over a finite coefficient field the generic maximum requires a separate attainment argument; the displayed maximum assumes an infinite field.
For weights (2, 3) and n = 7, the green triangle has area 49/12, while Π7 has area 7/2. There are 8 monomials and M7 = 7. When n is divisible by both weights, the triangle has integral vertices and Mn = n²/(wxwy).
Background: P. Mondal, How many zeroes?; for the translation argument, see G. Binyamini, §4.2, Remark 19.
Cutoff normalization, intrinsic degree, and the classical formula
Here n² is the product of the two common bounds. It equals γw(P)γw(Q) only when both ECO values equal n. The intrinsic ECO intersection degree instead uses each pair’s actual values:
= supdim k[x,y]/(P,Q)<∞ dimk k[x,y]/(P,Q) / (γw(P)γw(Q))
= 1/(wxwy).
The pair P = x, Q = y already attains the last value. For the bounded maximum, powers align both ECO values with n at common multiples of the weights.
With ordinary degree, dim Vn = (n + 1)(n + 2)/2 and Mn = n², attained by P = xn, Q = yn. This recovers the classical Hilbert polynomial of the plane: its leading coefficient is 1/2!, whereas the degree is 1. Equivalently, in k[x,y](x,y), the local Hilbert–Samuel function length(R/(x,y)n+1) has the same leading coefficient.
For weighted degree, the sublevels form a weighted filtration, generally not the powers of one ideal. The ECO theorem provides the corresponding growth–intersection statement. Here R is a finite-type integral domain, γw is finite everywhere, the finite-part rank is 1, and its self-degree is positive. These are the hypotheses relevant to this application.
ECO degree and Hilbert–Samuel formula: Robert Wilms’s manuscript on ECO functions, sections “The degree function of tuples of eco functions” and “Hilbert–Samuel formula”.
Rational numbers · logarithmic height
A small number can have a big height.
Write x = a/b in lowest terms, with b > 0. Its multiplicative height is H(x) = max{|a|, b}, and its usual logarithmic height is h(x) = log H(x). For example, 1/2 and 49/100 are close in value, but their heights are 2 and 100. We explore the truncated logarithmic height
For n ≥ 1, the bounded set is exactly
B = ⌊en⌋.
n = 2.7 · B = ⌊eⁿ⌋ = 14
Interactive range: 1 ≤ n ≤ 4. Use the slider, enter a value, or press + / −. Points enter when ⌊en⌋ increases.Each dot represents one reduced rational number: its horizontal coordinate is a/b and its vertical row is its denominator b. Thus 1/2 appears only on row b = 2; 2/4 adds no second point. The red marker selects a fraction; a hollow marker indicates a fraction outside the height bound.
Algebra → valuation points → convex geometry
What survives at large scale?
A valuation turns nonzero algebraic elements into lattice points. Rescale the points from the n-th sublevel by n: under the appropriate boundedness and rank hypotheses, a convex body records their leading growth.
For k[x,y], use the valuation given by the lexicographically least monomial exponent. The limiting body is the unit simplex. Its area gives the leading coefficient of the dimension polynomial; multiplying by 2! recovers the self-intersection of the line on P2.
Compare the rational-height experiment
For rational numbers, coprime numerators and denominators give a different counting problem. With B = ⌊en⌋,
log #η≤n / n → 2.
This describes the exponential growth of rational numbers of bounded logarithmic height. Since η fails the ECO addition axiom, it is not an application of the ECO volume formula.