Frigid Touch

No, units can stick around on the board with 0 (or less!) Might. They will die if they take any damage, but "0" is not a valid amount of damage, so it takes 1 damage to kill a unit with 0 or less Might. We realize that this is a little weird, but it's important to us that cards like Frigid Touch can't be used to actually kill units, only weaken them.

142.4.b. Lethal Damage for a Unit is a non-zero amount greater than or equal to that Unit's Might. Example: A unit has 5 [M] and 3 damage marked on it. Frigid Touch is played targeting that unit. When it resolves, the unit's Might becomes 3, and it will have lethal damage marked on it. Example: A unit has 0 [M]. In order to have lethal damage marked on it, it must have at least 1 damage marked on it.

355.14.h.1. That player cannot choose to have fewer Targets than they have damage to split when choosing which Targets cease being Targets. Example: A player plays Alpha Strike, which reads in part "Choose a friendly unit. It deals damage equal to its Might split among enemy units at battlefields." They target their 5 [M] unit and five 1 [M] enemy Recruits at battlefields. In reaction, their opponent plays Feral Strength targeting one of their recruits, and Frigid Touch targeting the 5 [M] unit. When Alpha Strike resolves, they have 3 damage to split, and must choose which targets cease being targets. They can't choose for 4 of the units to cease being targets so that they can deal 3 to the 3 [M] recruit. They can only choose at most 2 of the targets to cease being targets.

356.4.c. Discounts that only apply to a component of the cost will be applied when that component is added to the cost of the spell and before any other discounts. Example: Ezreal, Prodigy reads "optional additional costs you pay cost [1] or [A] less." When playing a Frigid Touch and choosing to pay the additional cost in step 2, as soon as the additional cost is added to the cost of the spell, Ezreal, Prodigy's discount is applied to it.