# What is the probability of 'yes' to this likely $coNP$ problem?

1. Pick a set of primitive (gcd of coordinates is $$1$$) integer points $$\mathcal T$$ in $$\mathbb Z^n$$.

2. Denote the set of $$n$$ many algebraically independent homogeneous system of polynomials (thus zero-dimensional system) in $$\mathbb Z[x_1,\dots,x_n]$$ with degree $$2$$ with absolute value of coefficients bound by $$2^{n^c}$$ size at a fixed $$c\geq1$$ with all the points in $$\mathcal T$$ being their primitive integer roots to be $$\mathcal P$$.

Note that for a system of polynomials $$H\in\mathcal P$$ we may have additional real roots in $$\mathbb R^n$$. However integer roots come from every point in $$\mathcal T$$ and are the only integer roots in $$H$$.

Note that for system of polynomials $$H_1,H_2\in\mathcal P$$ we have all the integer roots agreeing. However real roots may not agree and in fact may have no real roots in common.

Consider the problem 'Given $$H_1$$ and $$H_2$$ from $$\mathcal P$$ does the intersection of roots equal $$\mathcal T$$?'.

I would also think with probability $$1$$ we have answer is 'yes'.

1. What is the probability that the answer is 'yes'?

2. Is this in $$NP$$ (note it is not clear it is in $$coNP$$ either as we might need infinite precision real roots to certify 'no')?