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The Newton polygon

_Maxim_ 729
November 08 2017
2

Inspired by this question. There is a simple iterated procedure that can generate the Puiseux expansion.

Note the use of convert(..., rational, exact) to preserve the digits of the floating-point numbers.

> 

(proc () global COcrit, P; Digits := 30; COcrit := 1/convert(2*0.115e-12, rational, exact); P := (`@`(`@`(rcurry(unapply, [NO2, O3]), numer), rcurry(convert, rational, exact)))(1.8027350279544425625*O3^8/10^49+(2.982942852010948125*CO/10^49+2.27611508925183215625*NO2/10^47+3.754849807690565625/10^37)*O3^7+(1.2339511797887015625*CO^2/10^49+2.4836920553034140625*CO/10^37+(-1)*5.9759150186390484375*NO2/10^35+6.3862221928648528125*NO2^2/10^46+1.88311680046014890625*CO*NO2/10^47+(-1)*9.69704144499346875/10^24)*O3^6+((-1)*1.71193039685859375*CO^2/10^37+(-1)*5.7098636544065625*CO/10^24+(-1)*1.277325786277575*NO2/10^21+(-1)*1.0801570017944671875*NO2^2/10^32+(-1)*2.9081778565815421875*CO*NO2/10^34+(-1)*3.66453227489203125/10^11)*O3^5+(1.9152220505625*CO^2/10^25+8.1035862984375*CO/10^13+1.40846609345625*NO2/10^10+(-1)*5.19285353257125*NO2^2/10^21+(-1)*1.55036925507375*CO*NO2/10^21+(-1)*1.844759695120875*CO^2*NO2/10^34+(-1)*1.876842472427325*CO*NO2^2/10^32-7.201634275625)*O3^4+(1.793091274970625*NO2^2/10^7+8618.14231275*NO2+(-1)*2.298266460675*CO^2*NO2/10^22+(-1)*9.5902239009375*CO*NO2/10^10+(-1)*1.685705248305*CO*NO2^2/10^20+9.2666503797075*NO2^3/10^19)*O3^3+((-1)*2.5638555726*10^6*NO2^2+6.894799382025*NO2^2*CO^2/10^20+2.7563954788125*CO*NO2^2/10^7+3.5073544682475*NO2^3*CO/10^18+(-1)*0.604340578881e-4*NO2^3+(-1)*3.5683519372605*NO2^4/10^16)*O3^2+((-1)*8.75499651*10^6*NO2^3+0.482686765875e-5*NO2^3*CO+(-1)*0.98216263665e-4*NO2^4)*O3+5.4066609*10^7*NO2^4) end proc)()

We're interested in the asymptotics of RootOf(P(NO2, _Z)) for small NO2.

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plot3d(RootOf(P(NO2, _Z)), CO = .5*COcrit .. 2*COcrit, NO2 = 10^(-3) .. 10^(-6))

Start with P(NO2, Z) and look for the expansion for Z when NO2 is small.

Expand P(NO2, Z) into monomials and convert each monomial a*Typesetting:-mi("NO2", italic = "true", mathvariant = "italic")^Typesetting:-mi("p", italic = "true", mathvariant = "italic")*Typesetting:-mi("Z", italic = "true", mathvariant = "italic")^Typesetting:-mi("q", italic = "true", mathvariant = "italic") into the point [p, q].

> 

([op])(collect(P(NO2, Z), [NO2, Z], distributed, normal)); map(proc (e) options operator, arrow; `~`[degree](e, [NO2, Z]) end proc, %)

[[4, 2], [4, 1], [4, 0], [3, 3], [3, 2], [3, 1], [2, 6], [2, 5], [2, 4], [2, 3], [2, 2], [1, 7], [1, 6], [1, 5], [1, 4], [1, 3], [0, 8], [0, 7], [0, 6], [0, 5], [0, 4]]

 (1)

The Newton polygon is the convex hull of the points.

> 

newtonPoly := proc (P, xy) local terms, pts; terms := ([op])(collect(P, xy, distributed, normal)); pts := map(proc (e) options operator, arrow; `~`[degree](e, xy) end proc, terms); plots:-display(plottools:-polygon(simplex:-convexhull(pts)), plottools:-point(pts), symbol = solidcircle, symbolsize = 15, style = line, thickness = 3, color = [magenta, black], axis = [gridlines = spacing(1)]) end proc
> 

newtonPoly(P(NO2, Z), [NO2, Z])

 

The side closest to the origin will correspond to the asymptotics for small NO2.

Sum the corresponding monomials and solve for Z.

> 

sideAsympt := proc (P, xy, pts) local lead; lead := add(coeff(coeff(P, xy[1], p[1]), xy[2], p[2])*xy[1]^p[1]*xy[2]^p[2], `in`(p, pts)); ([solve])(lead, xy[2]) end proc
> 

sideAsympt(P(NO2, Z), [NO2, Z], [[0, 4], [1, 3], [2, 2], [3, 1], [4, 0]])

[600*NO2, 600*NO2, -57000000000000000*NO2/(4071*CO-17795000000000000), -228000000000000000*NO2/(4071*CO+35020000000000000)]

 (2)

We have obtained the first term in the expansion. If CO>COcrit, the smallest positive root is the one asymptotic to 600*NO2. That will be the value of RootOf(P(NO2, _Z)).

The expansion can be continued in the same manner.

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newtonPoly(P(NO2, NO2*(600+Z)), [NO2, Z])

 

In this case we don't have to choose between sides with different slopes. (Compare to "P(NO2,600*NO2+Z).)"

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sideAsympt(P(NO2, NO2*(600+Z)), [NO2, Z], [[4, 2], [5, 0]]); `assuming`([simplify(`assuming`([simplify(%)], [NO2 > 0]))], [CO > COcrit])

[(531/430000)*354^(1/2)*NO2^(1/2)*(200000000000000+23*CO)^(1/2)/(23*CO-100000000000000)^(1/2), -(531/430000)*354^(1/2)*NO2^(1/2)*(200000000000000+23*CO)^(1/2)/(23*CO-100000000000000)^(1/2)]

 (3)

We get `~`[Z]*NO2^(1/2), so the next order in the expansion is NO2^(3/2).

Again, if we want the solution that corresponds to RootOf(P(NO2, _Z)), we should take the negative term, as the principal root will be the smaller one.

Find one more term. To avoid fractional powers, take NO2r = NO2^(1/2).

> 

approx := 600*NO2r^2+NO2r^3*(-(531/430000)*sqrt(354)*sqrt(200000000000000+23*CO)/sqrt(23*CO-100000000000000)+Z)
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newtonPoly(P(NO2r^2, approx), [NO2r, Z])

 
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sideAsympt(P(NO2r^2, approx), [NO2r, Z], [[10, 1], [11, 0]])

[(4779/36980000000000)*(4341503*CO^2+29548445000000000000*CO-209981000000000000000000000000000)*NO2r/(529*CO^2-4600000000000000*CO+10000000000000000000000000000)]

 (4)

We get `~`[Z]*NO2r, so we have obtained the coefficient at NO2r^3*NO2r = NO2^2.

All of this can be done using the algcurves:-puiseux command. The only difficulty is the unwieldy expressions that algcurves:-puiseux will generate for P.

Let's also find the NO2^2 term by the method of dominant balance. Suppose that we don't know p yet and are looking for the term alpha*NO2^p = alpha*NO2p.

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approx := 600*NO2r^2-(531/430000)*sqrt(354)*sqrt(200000000000000+23*CO)*NO2r^3/sqrt(23*CO-100000000000000)+alpha*NO2p
> 

terms := (`@`([op], collect))(P(NO2r^2, approx), [NO2r, NO2p], distributed, normal)

Find all exponents of NO2, coming from NO2r and from NO2p.

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pows := map(proc (t) options operator, arrow; (1/2)*degree(t, NO2r)+p*degree(t, NO2p) end proc, terms)
> 

plot(pows, p = 1 .. 2, view = 4 .. 6, annotation = pows)

 

At p=2, the two dominant terms NO2^(11/2) and NO2^(7/2+p) can balance each other.

> 

(`@`(normal, coeff))(subs(NO2p = NO2r^4, P(NO2r^2, approx)), NO2r, 11); ([solve])(%, alpha)

[(4779/36980000000000)*(4341503*CO^2+29548445000000000000*CO-209981000000000000000000000000000)/(529*CO^2-4600000000000000*CO+10000000000000000000000000000)]

 (5)
> 

``

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