Alright, so I need to find the area of the convex polygon formed by the solutions of the equation \( z^4 + 4z^3i - 6z^2 - 4zi - i = 0 \). The area is supposed to be in the form \( p^{a/b} \) where \( p \) is prime, and \( a \) and \( b \) are coprime. Then I need to find \( a + b + p \). Hmm, okay. Let's start by understanding the problem step by step.

First, the equation is a quartic (degree 4) polynomial in \( z \). The solutions are the roots of this polynomial, which are complex numbers. Each root corresponds to a vertex in the complex plane. So, these four roots form a convex polygon, and I need to compute its area.

Since the polynomial is of degree 4, there are four roots, so the polygon is a quadrilateral. To find the area of this quadrilateral, I need to know the coordinates of each vertex in the complex plane. But maybe there's a smarter way without finding each root explicitly?

Alternatively, perhaps there's a pattern or symmetry in the roots that can help me determine the shape of the quadrilateral and hence its area. Let me think.

First, let's analyze the polynomial: \( z^4 + 4z^3i - 6z^2 - 4zi - i = 0 \). It might be helpful to factor this polynomial or recognize it as a transformation of a simpler polynomial.

Notice that the coefficients involve \( i \), which is the imaginary unit. It might be useful to substitute \( w = z \cdot i \) or something similar to simplify the equation. Let me try substituting \( w = z \cdot i \). Let's see:

If \( w = iz \), then \( z = -i w \). Let's substitute \( z = -i w \) into the equation:

\( (-i w)^4 + 4(-i w)^3 i - 6(-i w)^2 - 4(-i w)i - i = 0 \).

Let's compute each term step by step:

1. \( (-i w)^4 = (-i)^4 w^4 = (1) w^4 = w^4 \) because \( (-i)^4 = (-i)^2 \cdot (-i)^2 = (-1)^2 \cdot i^2 \cdot (-1)^2 \cdot i^2 = 1 \cdot (-1) \cdot 1 \cdot (-1) = 1 \).

2. \( 4(-i w)^3 i = 4(-i)^3 w^3 i = 4(-i)^3 i w^3 \). Let's compute \( (-i)^3 \): \( (-i)^3 = (-i) \cdot (-i) \cdot (-i) = (-i)^2 \cdot (-i) = (-1)^2 i^2 \cdot (-i) = 1 \cdot (-1) \cdot (-i) = i \). So, this term becomes \( 4 \cdot i \cdot i \cdot w^3 = 4i^2 w^3 = 4(-1) w^3 = -4w^3 \).

3. \( -6(-i w)^2 = -6(-i)^2 w^2 = -6(-1)^2 i^2 w^2 = -6(1)(-1) w^2 = 6w^2 \).

4. \( -4(-i w)i = -4(-i)i w = 4i^2 w = 4(-1)w = -4w \).

5. The last term is \( -i \).

Putting it all together:

\( w^4 - 4w^3 + 6w^2 - 4w - i = 0 \).

Hmm, that's interesting. Now, the equation is \( w^4 - 4w^3 + 6w^2 - 4w - i = 0 \). Wait, the first four terms look familiar. Let's check:

\( w^4 - 4w^3 + 6w^2 - 4w \). That's the expansion of \( (w - 1)^4 \) minus the constant term. Let's verify:

\( (w - 1)^4 = w^4 - 4w^3 + 6w^2 - 4w + 1 \). So, subtracting 1 gives \( w^4 - 4w^3 + 6w^2 - 4w \). Therefore, the equation becomes:

\( (w - 1)^4 - 1 - i = 0 \).

So, \( (w - 1)^4 = 1 + i \).

Therefore, \( (w - 1)^4 = 1 + i \). So, to solve for \( w \), we take the fourth roots of \( 1 + i \). Then, \( w = 1 + \sqrt[4]{1 + i} \). Therefore, the roots \( w \) are given by \( w = 1 + \sqrt[4]{1 + i} \cdot e^{2\pi i k /4} \) for \( k = 0, 1, 2, 3 \).

But since \( w = iz \), then \( z = -i w = -i (1 + \sqrt[4]{1 + i} \cdot e^{\pi i k /2}) \).

Therefore, each root \( z \) is \( -i \) times \( 1 + \) a fourth root of \( 1 + i \). Hmm, this seems a bit complicated. Maybe there's another way to express \( 1 + i \) in polar form?

Yes, \( 1 + i \) can be written as \( \sqrt{2} e^{i\pi/4} \). Therefore, the fourth roots of \( 1 + i \) are \( (\sqrt{2})^{1/4} e^{i(\pi/16 + \pi k /2)} \), for \( k = 0, 1, 2, 3 \). Let's compute \( (\sqrt{2})^{1/4} = (2^{1/2})^{1/4} = 2^{1/8} \).

Therefore, the fourth roots are \( 2^{1/8} e^{i(\pi/16 + 2\pi k /8)} \), since \( \pi k /2 = 2\pi k /8 \). Wait, \( k = 0, 1, 2, 3 \). So, angles are \( \pi/16, 9\pi/16, 17\pi/16, 25\pi/16 \).

Therefore, each \( w \) is \( 1 + 2^{1/8} e^{i(\pi/16 + 2\pi k /8)} \). Then, \( z = -i w = -i (1 + 2^{1/8} e^{i(\pi/16 + 2\pi k /8)}) \).

Therefore, the four roots \( z \) are:

\( z_k = -i \left(1 + 2^{1/8} e^{i(\pi/16 + \pi k /2)} \right) \) for \( k = 0, 1, 2, 3 \).

Hmm. So, each \( z_k \) is a complex number with a real part and an imaginary part. Let me try to write each \( z_k \) in rectangular form (a + bi) to find their coordinates.

First, let's compute \( -i \times e^{i\theta} \). Remember that multiplying by \( -i \) is equivalent to rotating by \( -\pi/2 \) radians (clockwise by 90 degrees). So, \( -i e^{i\theta} = e^{-i\pi/2} e^{i\theta} = e^{i(\theta - \pi/2)} \). Therefore, \( -i \times \text{something} \) is just that something rotated by \( -\pi/2 \).

Therefore, \( z_k = -i \times 1 -i \times 2^{1/8} e^{i(\pi/16 + \pi k /2)} \).

Wait, but \( -i \times 1 = -i \), and \( -i \times 2^{1/8} e^{i(\pi/16 + \pi k /2)} = 2^{1/8} e^{i(\pi/16 + \pi k /2 - \pi/2)} = 2^{1/8} e^{i(\pi/16 - \pi/2 + \pi k /2)} = 2^{1/8} e^{i(-7\pi/16 + \pi k /2)} \).

Therefore, each \( z_k \) can be written as:

\( z_k = -i + 2^{1/8} e^{i(-7\pi/16 + \pi k /2)} \).

Alternatively, maybe it's better to compute each \( z_k \) directly.

Let me attempt to compute each \( z_k \) for \( k = 0, 1, 2, 3 \):

For \( k = 0 \):

\( z_0 = -i \left(1 + 2^{1/8} e^{i\pi/16} \right) = -i - i 2^{1/8} e^{i\pi/16} \).

Similarly, for \( k = 1 \):

\( z_1 = -i \left(1 + 2^{1/8} e^{i(9\pi/16)} \right) = -i - i 2^{1/8} e^{i(9\pi/16)} \).

For \( k = 2 \):

\( z_2 = -i \left(1 + 2^{1/8} e^{i(17\pi/16)} \right) = -i - i 2^{1/8} e^{i(17\pi/16)} \).

For \( k = 3 \):

\( z_3 = -i \left(1 + 2^{1/8} e^{i(25\pi/16)} \right) = -i - i 2^{1/8} e^{i(25\pi/16)} \).

So, each \( z_k \) is the sum of two complex numbers: \( -i \) and another term. Let's try to express each \( z_k \) in rectangular coordinates.

First, note that \( e^{i\theta} = \cos\theta + i\sin\theta \). Therefore, multiplying by \( -i \):

\( -i e^{i\theta} = -i (\cos\theta + i\sin\theta) = -i\cos\theta - i^2 \sin\theta = -i\cos\theta + \sin\theta \).

So, \( -i e^{i\theta} = \sin\theta - i\cos\theta \).

Therefore, each \( z_k \) can be written as:

\( z_k = -i + \sin\theta_k - i\cos\theta_k \), where \( \theta_k = \pi/16 + \pi k /2 \).

Therefore, \( z_k = (-i) + (\sin\theta_k - i\cos\theta_k) = (\sin\theta_k) + i(-1 - \cos\theta_k) \).

Therefore, the rectangular coordinates of each \( z_k \) are \( (\sin\theta_k, -1 - \cos\theta_k) \).

So, the four vertices are:

1. \( z_0 \): \( (\sin(\pi/16), -1 - \cos(\pi/16)) \)
2. \( z_1 \): \( (\sin(9\pi/16), -1 - \cos(9\pi/16)) \)
3. \( z_2 \): \( (\sin(17\pi/16), -1 - \cos(17\pi/16)) \)
4. \( z_3 \): \( (\sin(25\pi/16), -1 - \cos(25\pi/16)) \)

Now, let's compute these coordinates numerically to see if they form a symmetric shape.

But before that, maybe there's a way to see the polygon's symmetry. Let's note that the angles \( \theta_k \) for \( k = 0, 1, 2, 3 \) are \( \pi/16, 9\pi/16, 17\pi/16, 25\pi/16 \). Each subsequent angle is increased by \( \pi/2 \) (since \( 9\pi/16 - \pi/16 = \pi/2 \), etc.). Therefore, the four points are equally spaced around a circle with radius \( 2^{1/8} \), but shifted by some angle.

But since each \( z_k \) is \( -i \times (1 + r e^{i\theta}) \), where \( r = 2^{1/8} \), the points are all located at a distance of \( |1 + r e^{i\theta}| \) from the origin. However, since \( 1 \) and \( r e^{i\theta} \) are vectors, their sum's magnitude depends on the angle between them.

But maybe instead of computing coordinates, there's a better approach.

Wait, since all four roots are related by rotations of \( \pi/2 \), perhaps the polygon is a square? But in the complex plane, if you have four points each rotated by 90 degrees, they might form a square. However, in this case, each root is \( -i \times (1 + \text{fourth root of } 1 + i) \), so the points are scaled and rotated versions of the fourth roots of \( 1 + i \), shifted by \( -i \).

Alternatively, maybe the polygon is a regular quadrilateral, but given the nature of the roots, it might not be regular. Let's check the coordinates.

Let me compute the coordinates for each \( z_k \):

First, compute \( \theta_0 = \pi/16 \approx 11.25^\circ \)

\( \sin(\pi/16) \approx 0.1951 \)

\( \cos(\pi/16) \approx 0.980785 \)

Therefore, \( z_0 \) coordinates: \( (0.1951, -1 - 0.9808) = (0.1951, -1.9808) \)

Similarly, \( \theta_1 = 9\pi/16 \approx 101.25^\circ \)

\( \sin(9\pi/16) \approx 0.980785 \)

\( \cos(9\pi/16) \approx -0.1951 \)

Therefore, \( z_1 \) coordinates: \( (0.9808, -1 - (-0.1951)) = (0.9808, -0.8049) \)

Next, \( \theta_2 = 17\pi/16 \approx 181.25^\circ \)

\( \sin(17\pi/16) \approx -0.1951 \)

\( \cos(17\pi/16) \approx -0.980785 \)

Therefore, \( z_2 \) coordinates: \( (-0.1951, -1 - (-0.9808)) = (-0.1951, -0.0192) \)

Finally, \( \theta_3 = 25\pi/16 \approx 281.25^\circ \)

\( \sin(25\pi/16) \approx -0.980785 \)

\( \cos(25\pi/16) \approx 0.1951 \)

Therefore, \( z_3 \) coordinates: \( (-0.9808, -1 - 0.1951) = (-0.9808, -1.1951) \)

So, the four vertices are approximately:

1. \( (0.1951, -1.9808) \)
2. \( (0.9808, -0.8049) \)
3. \( (-0.1951, -0.0192) \)
4. \( (-0.9808, -1.1951) \)

Plotting these points, they seem to form a quadrilateral that's symmetric about the y-axis. Let's check the coordinates:

Looking at the x-coordinates: 0.1951, 0.9808, -0.1951, -0.9808. The points are symmetric with respect to the y-axis.

Similarly, the y-coordinates: -1.9808, -0.8049, -0.0192, -1.1951. The two points on the left and right have similar y-values but mirrored.

Therefore, the quadrilateral is symmetric about the y-axis. Therefore, the area can be computed by finding the area of one half and doubling it.

Alternatively, since it's symmetric, maybe it's a trapezoid or a kite. Let's see.

Looking at the coordinates, points 1 and 2 are on the right side, points 3 and 4 are on the left. Wait, actually, points 1 and 4 are on the left and right, but their y-coordinates are different. Wait, maybe the quadrilateral is a parallelogram?

Wait, but with the given coordinates, let's compute the vectors between the points to see if they are parallel.

Compute vectors:

From point 1 to point 2: (0.9808 - 0.1951, -0.8049 - (-1.9808)) = (0.7857, 1.1759)

From point 2 to point 3: (-0.1951 - 0.9808, -0.0192 - (-0.8049)) = (-1.1759, 0.7857)

From point 3 to point 4: (-0.9808 - (-0.1951), -1.1951 - (-0.0192)) = (-0.7857, -1.1759)

From point 4 to point 1: (0.1951 - (-0.9808), -1.9808 - (-1.1951)) = (1.1759, -0.7857)

Looking at these vectors: the first vector is (0.7857, 1.1759), the second is (-1.1759, 0.7857), the third is (-0.7857, -1.1759), and the fourth is (1.1759, -0.7857). If you look at the first and third vectors: (-1.1759, 0.7857) is negative of the second vector, and (0.7857, 1.1759) is the first vector. Similarly, the vectors are rotated by 90 degrees. Wait, if you rotate (0.7857, 1.1759) by 90 degrees, you get (-1.1759, 0.7857), which is the second vector. Similarly, the third vector is (-0.7857, -1.1759), which is the negative of the first vector. The fourth vector is (1.1759, -0.7857), which is the negative of the second vector. Therefore, the sides are equal in length and adjacent sides are perpendicular. Therefore, the quadrilateral is a square? Wait, but with these vectors, the sides are equal and the angles between them are 90 degrees. Therefore, yes, the quadrilateral is a square.

Wait, but looking at the coordinates, the side lengths are:

From point 1 to point 2: sqrt(0.7857^2 + 1.1759^2) ≈ sqrt(0.617 + 1.383) ≈ sqrt(2) ≈ 1.414.

Similarly, from point 2 to point 3: sqrt((-1.1759)^2 + 0.7857^2) ≈ sqrt(1.383 + 0.617) ≈ sqrt(2) ≈ 1.414.

Same for the other sides. Therefore, all sides are equal in length. Moreover, adjacent sides are perpendicular. Therefore, the quadrilateral is a square. Therefore, the area is side length squared. Wait, but side length is sqrt(2), so area would be 2. But in the problem, it's supposed to be expressed as \( p^{a/b} \). But 2 is 2^1, so p=2, a=1, b=1, so a+b+p=4. But that seems too easy, and maybe I made a mistake here.

Wait, but hold on. Let's check the coordinates again. Maybe they don't form a square. Let's compute the distances between consecutive points.

Distance between z0 and z1:

Coordinates:

z0: (0.1951, -1.9808)

z1: (0.9808, -0.8049)

Distance squared: (0.9808 - 0.1951)^2 + (-0.8049 + 1.9808)^2 ≈ (0.7857)^2 + (1.1759)^2 ≈ 0.617 + 1.383 ≈ 2. So distance is sqrt(2).

Similarly, distance between z1 and z2:

z1: (0.9808, -0.8049)

z2: (-0.1951, -0.0192)

Distance squared: (-0.1951 - 0.9808)^2 + (-0.0192 + 0.8049)^2 ≈ (-1.1759)^2 + (0.7857)^2 ≈ 1.383 + 0.617 ≈ 2. Distance sqrt(2).

Similarly, distance between z2 and z3:

z2: (-0.1951, -0.0192)

z3: (-0.9808, -1.1951)

Distance squared: (-0.9808 + 0.1951)^2 + (-1.1951 + 0.0192)^2 ≈ (-0.7857)^2 + (-1.1759)^2 ≈ 0.617 + 1.383 ≈ 2. Distance sqrt(2).

Distance between z3 and z0:

z3: (-0.9808, -1.1951)

z0: (0.1951, -1.9808)

Distance squared: (0.1951 + 0.9808)^2 + (-1.9808 + 1.1951)^2 ≈ (1.1759)^2 + (-0.7857)^2 ≈ 1.383 + 0.617 ≈ 2. Distance sqrt(2).

Therefore, all sides are equal. Now, check the angles. Since the vectors between consecutive points are rotated by 90 degrees, as previously observed, the angles between consecutive sides are 90 degrees. Therefore, the quadrilateral is indeed a square with side length sqrt(2). Therefore, its area is (sqrt(2))^2 = 2. But 2 can be written as 2^{1}, so p=2, a=1, b=1, so a+b+p=4.

Wait, but the problem states "the area of this polygon can be expressed in the form \( p^{a/b} \)", and 2 is already an integer, so 2 = 2^{1/1}. So, p=2, a=1, b=1, so a+b+p=4. But that seems too straightforward. Maybe I made a mistake in assuming the quadrilateral is a square?

Wait, let's check the coordinates again. Wait, the coordinates are approximate. Let's compute more precisely.

Compute the coordinates symbolically:

Let me denote \( r = 2^{1/8} \), so \( r^8 = 2 \). Then, \( 1 + r e^{i\theta} \) is the vector from the origin to the point \( (1 + r \cos\theta, r \sin\theta) \). Then, multiplying by \( -i \) is equivalent to rotating this vector by -90 degrees (clockwise) and scaling by -1 (which is equivalent to reflection over the real axis and rotation by 180 degrees). Wait, perhaps another way.

Alternatively, since \( z = -i w \), and \( w = 1 + \sqrt[4]{1 + i} e^{2\pi i k /4} \), then each \( z \) is a rotation and scaling of the fourth roots of \( 1 + i \). However, maybe this is complicating things.

Alternatively, since we have the coordinates approximately, but when we computed the side lengths, all sides were sqrt(2), and the angles between sides are 90 degrees, so it's a square. Therefore, area is 2. Therefore, the answer would be 4. However, maybe I miscalculated.

Wait, let's check the coordinates again with more precision.

Wait, the coordinates are:

z0: (sin(π/16), -1 - cos(π/16))

z1: (sin(9π/16), -1 - cos(9π/16))

z2: (sin(17π/16), -1 - cos(17π/16))

z3: (sin(25π/16), -1 - cos(25π/16))

Compute these exactly:

First, note that sin(π/16) and sin(9π/16) and others can be expressed in terms of radicals, but maybe it's better to compute exact values.

But perhaps a better approach is to compute the vectors between the points and check if they are equal in length and perpendicular.

But in the approximate coordinates, we saw that all sides are sqrt(2), and adjacent sides have dot product zero (since they are rotated by 90 degrees). Therefore, the quadrilateral is a square.

Alternatively, perhaps there's a miscalculation in coordinates. Let me re-examine.

Wait, if we look at the coordinates:

z0: (sin(π/16), -1 - cos(π/16)) ≈ (0.1951, -1.9808)

z1: (sin(9π/16), -1 - cos(9π/16)) ≈ (0.9808, -0.8049)

z2: (sin(17π/16), -1 - cos(17π/16)) ≈ (-0.1951, -0.0192)

z3: (sin(25π/16), -1 - cos(25π/16)) ≈ (-0.9808, -1.1951)

Plotting these points, they seem to form a square rotated by 45 degrees relative to the axes.

Wait, for example, the distance between z0 and z3 is sqrt[ ( -0.9808 - 0.1951 )^2 + ( -1.1951 + 1.9808 )^2 ] ≈ sqrt[ (-1.1759)^2 + (0.7857)^2 ] ≈ sqrt[1.383 + 0.617] ≈ sqrt[2] ≈ 1.414. Similarly, the distance between z1 and z2 is sqrt[ (-1.1759)^2 + (0.7857)^2 ] ≈ sqrt[2]. So, all sides are sqrt(2). The diagonals: distance between z0 and z2 is sqrt[ (-0.1951 - 0.1951)^2 + (-0.0192 + 1.9808)^2 ] ≈ sqrt[ (-0.3902)^2 + (1.9616)^2 ] ≈ sqrt[0.1522 + 3.847] ≈ sqrt[4] = 2. Similarly, the distance between z1 and z3 is sqrt[ (-0.9808 - 0.9808)^2 + (-1.1951 + 0.8049)^2 ] ≈ sqrt[ (-1.9616)^2 + (-0.3902)^2 ] ≈ sqrt[3.847 + 0.1522] ≈ sqrt[4] = 2. Therefore, the diagonals are 2. In a square, the area is (diagonal^2)/2 = 4/2 = 2. Therefore, the area is indeed 2. Therefore, the answer would be 2^{1}, so p=2, a=1, b=1, and a+b+p=4.

But the problem states "the area can be expressed in the form \( p^{a/b} \)", and 2 is an integer, so 2 = 2^{1/1}, so a=1, b=1. Therefore, the answer is 4. But is this correct?

Wait, but maybe there's a miscalculation in the assumption that the polygon is a square. Let me verify again.

The four points:

z0: (sin(π/16), -1 - cos(π/16)) ≈ (0.1951, -1.9808)

z1: (sin(9π/16), -1 - cos(9π/16)) ≈ (0.9808, -0.8049)

z2: (sin(17π/16), -1 - cos(17π/16)) ≈ (-0.1951, -0.0192)

z3: (sin(25π/16), -1 - cos(25π/16)) ≈ (-0.9808, -1.1951)

Plotting these points, they look like a square rotated by 45 degrees, with side length sqrt(2), and diagonals of length 2. Therefore, the area is 2. Therefore, the answer is 2^{1}, so p=2, a=1, b=1. Therefore, a+b+p=4.

But wait, maybe there's a mistake in the substitution. Let me check the substitution again.

Original equation: \( z^4 + 4z^3i -6z^2 -4zi -i=0 \).

Substituted \( w = iz \), so \( z = -i w \).

Then the equation becomes \( w^4 -4w^3 +6w^2 -4w -i =0 \), which we factored as \( (w -1)^4 =1 +i \).

Therefore, \( (w -1)^4 =1 +i \), so \( w =1 + \sqrt[4]{1 +i} \). Then, since \( 1 +i \) is a complex number, its fourth roots are the four solutions. Therefore, \( w =1 + \sqrt[4]{1 +i} e^{2\pi i k /4} \), for k=0,1,2,3. Then, \( z = -i w = -i (1 + \sqrt[4]{1 +i} e^{2\pi i k /4}) \).

Therefore, the coordinates are correct. Then, when computing the coordinates in rectangular form, they are as above.

Therefore, the conclusion is that the area is 2. Therefore, the answer is 4.

But the problem says "The area of this polygon can be expressed in the form \( p^{a/b} \)", which 2 is in the form \( 2^{1/1} \), so p=2, a=1, b=1, which gives a+b+p=4.

But maybe there's a mistake in assuming the polygon is a square. Wait, perhaps the polygon is not a square but another quadrilateral? Wait, according to the coordinates, the polygon is symmetric about the y-axis. Let's check the coordinates again:

z0: (0.1951, -1.9808)

z1: (0.9808, -0.8049)

z2: (-0.1951, -0.0192)

z3: (-0.9808, -1.1951)

Plotting these points, they form a quadrilateral with two points on the right side (z0 and z1), two points on the left (z2 and z3). The top point is z2, which is near the origin, and the bottom points are z0 and z3. The sides between z0 and z1, z1 and z2, z2 and z3, z3 and z0.

But when connecting these points in order, z0, z1, z2, z3, the shape is a convex quadrilateral. Wait, but according to the coordinates, z0 is (0.1951, -1.9808), which is the lowest point on the right, then moving to z1: (0.9808, -0.8049), which is higher, then to z2: (-0.1951, -0.0192), which is near the origin, then to z3: (-0.9808, -1.1951), which is lower left, then back to z0. So, connecting these points in order, it's a trapezoid or a kite? Wait, actually, the sides between z0 and z1 is a diagonal going up, z1 to z2 is a diagonal going left and up, z2 to z3 is a diagonal going left and down, z3 to z0 is a diagonal going right and down. Hmm, perhaps it's a rectangle? But in our earlier calculation, we found all sides are sqrt(2), and adjacent sides are perpendicular. Therefore, it's a square.

But in reality, when we plot these points, they might not form a square. Wait, let's compute the vectors again.

From z0 to z1: (0.9808 - 0.1951, -0.8049 - (-1.9808)) = (0.7857, 1.1759)

From z1 to z2: (-0.1951 - 0.9808, -0.0192 - (-0.8049)) = (-1.1759, 0.7857)

From z2 to z3: (-0.9808 - (-0.1951), -1.1951 - (-0.0192)) = (-0.7857, -1.1759)

From z3 to z0: (0.1951 - (-0.9808), -1.9808 - (-1.1951)) = (1.1759, -0.7857)

So, the vectors are:

v1 = (0.7857, 1.1759)

v2 = (-1.1759, 0.7857)

v3 = (-0.7857, -1.1759)

v4 = (1.1759, -0.7857)

Each vector is the previous vector rotated by 90 degrees. For example, v2 is v1 rotated by 90 degrees clockwise (since x becomes y, y becomes -x). Similarly, v3 is v2 rotated by 90 degrees clockwise, etc. Therefore, the quadrilateral is a square with sides equal and angles 90 degrees. Therefore, the area is (sqrt(2))^2 = 2.

But why does the problem mention expressing it in the form \( p^{a/b} \)? Because 2 is already an integer, so \( 2 = 2^{1} \), which is \( 2^{1/1} \). So, p=2, a=1, b=1, so a+b+p=4. Therefore, the answer is 4.

However, I need to verify once again. Let's compute the area using coordinates. The shoelace formula might help here.

Using the shoelace formula for the coordinates:

List the coordinates in order:

z0: (0.1951, -1.9808)

z1: (0.9808, -0.8049)

z2: (-0.1951, -0.0192)

z3: (-0.9808, -1.1951)

Back to z0: (0.1951, -1.9808)

Compute the sum of x_i y_{i+1}:

(0.1951)(-0.8049) + (0.9808)(-0.0192) + (-0.1951)(-1.1951) + (-0.9808)(-1.9808)

= (-0.1569) + (-0.019) + (0.233) + (1.941)

≈ (-0.1569 - 0.019) + (0.233 + 1.941)

≈ (-0.1759) + (2.174)

≈ 1.9981

Compute the sum of y_i x_{i+1}:

(-1.9808)(0.9808) + (-0.8049)(-0.1951) + (-0.0192)(-0.9808) + (-1.1951)(0.1951)

= (-1.941) + (0.1569) + (0.0187) + (-0.233)

≈ (-1.941 + 0.1569) + (0.0187 - 0.233)

≈ (-1.7841) + (-0.2143)

≈ -2.0

Therefore, the shoelace area is 0.5 * |1.9981 - (-2.0)| = 0.5 * |3.9981| ≈ 1.999, which is approximately 2. Therefore, the area is indeed 2. Therefore, the answer is 2, which is 2^{1}, so a=1, b=1, p=2. Therefore, a+b+p=4.

But the problem says "the area of this polygon can be expressed in the form \( p^{a/b} \)", which 2 can be written as \( 2^{1/1} \), so the answer is 4.

But maybe I made a wrong substitution. Let me check the substitution again.

Original equation: \( z^4 +4z^3i -6z^2 -4zi -i =0 \). Let w = iz, so z = -i w.

Substitute into equation:

\( (-i w)^4 +4(-i w)^3 i -6(-i w)^2 -4(-i w)i -i =0 \)

Compute each term:

First term: \( (-i w)^4 = (i^4 w^4) = (1)w^4 = w^4 \)

Second term: \(4(-i w)^3 i = 4*(-i)^3 w^3 * i = 4*(i^3) w^3 * i = 4*(-i) w^3 * i = 4*(-i^2) w^3 = 4*1 w^3 =4w^3 \)

Third term: \( -6(-i w)^2 = -6*(i^2 w^2) = -6*(-1)w^2 =6w^2 \)

Fourth term: \( -4(-i w)i = -4*(-i^2)w = -4*(1)w = -4w \)

Fifth term: -i

Therefore, the equation becomes:

\( w^4 + 4w^3 +6w^2 -4w -i =0 \)

Which factors as \( (w -1)^4 =1 +i \). That seems correct. Therefore, substitution is correct.

Then, solving for w: \( (w -1)^4 =1 +i \). So, fourth roots of \(1 +i\). Therefore, the coordinates of z are as computed.

Therefore, the conclusion is that the area is 2, so the answer is 4. But since the problem mentions "the area of this polygon can be expressed in the form \( p^{a/b} \)", and 2 is already in that form with a=1, b=1, p=2, then the answer is 4.

But maybe there's a different approach where the area is expressed as a different power?

Wait, perhaps the square has sides of length sqrt(2), so area 2. But 2 can be written as 2^{1}, which is p=2, a=1, b=1. So that's correct. Therefore, the answer is 4.

**Final Answer**
\boxed{4}
Therefore, the area of the convex polygon formed by the solutions of the given equation can be expressed in the form \(2^1\), which gives \(p = 2\), \(a = 1\), and \(b = 1\). Therefore, the sum \(a + b + p\) is \(1 + 1 + 2 = 4\).
</think>